Longterm existence of solutions of a reaction diﬀusion system with non-local terms modeling an immune response - an interpretation-orientated proof

This paper shows the global existence and boundedness of solutions of a reaction diﬀusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local eﬀects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial diﬀerential equations are textbook examples in the ﬁeld, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time. We show the boundedness of the solution in the L 1 (Ω)- and the L 2 (Ω)- norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the L ∞ (Ω)-norm has practical relevance: Large immune responses lead to strong inﬂammations of the liver tissue. Strong inﬂammations negatively impact the health of an infected person and lead to grave secondary diseases. The


Introduction
Modeling the coupled dynamics of virus and the immune system during a liver infection caused by a hepatitis virus is challenging because the mechanisms behind persisting infections over month or years are still unknown. 1An opportunity for overcoming the problem of unknown mechanisms on the cell scale contains two integrative changes.First, the modeling scale is changed from the cell scale towards a mesoscopic scale on the length scale of a few centimeters.Second, the mechanisms, which are unknown in detail, are replaced by integrative mechanisms representing the commonly accepted properties of the unknown mechanisms.This change of view results in a compact model of partial differential equations.
Modeling inflammations with differential equations is a widely used approach.For example in Refs.2, 3, atherogenesis as a particular inflammation is modeled with reaction diffusion equations.In Ref. 2, instable states are interpreted as persisting infections, whereas in Ref. 3  traveling waves are interpreted as persisting infections.Reaction diffusion systems for modeling the dynamics of liver infections are presented in Refs.4, 5.In Refs.6-8 systems of ordinary equations are used for modeling the total amount of immune system cells and virus during a hepatitis C liver infection.
In Refs.9-13 liver inflammations are modeled by using reaction diffusion equations describing the virus concentration and the T cell population during an infection.As a specific feature, the reaction diffusion equations include a space-dependent and non-local term describing the inflow of T cells in a small part of the modeled region.The amount of inflowing T cells depends on the total virus amount in the regarded part of the liver.The dependency on the total virus amount is represented by an integral term over the whole domain.The non-local term models the T cell dispersal starting in the lymphs.
The description of the inflow region, called portal field, reflects some important parts of the real liver structure.Therefore, the term is desirable and necessary for modeling liver infections even if it makes the mathematical analysis of the model more difficult.One challenging task caused by the non-local and space-depending inflow term is the proof of the longterm existence of a solution.Often used results for parabolic partial differential equations are based on Lipschitz continuous reaction functions with respect to the state variable or require monotonous reaction functions.Due to the integral term and the oppositely acting mechanisms, these results are not directly applicable to the system modeling the dynamics of liver infections, see Section 3.  In this paper, the longterm existence and boundedness of solutions of the model proposed in Ref. 9 is proven and the results are interpreted https://doi.org/10.1016/j.padiff.2022.100446 in the light of the application.The focus therefore lies not only on adapting established theorems but on finding interpretable estimations on the way to an existence result.Therefore, the model is presented in Section 2. An important property of the reaction functions are the oppositely acting mechanisms like in the classical Lotka-Volterra equations and in nearly all population dynamics models.The nonlocal term is a new feature compared to the classical model and influences the dynamics of the model much more than only by its position-dependency.
In Section 3, the longterm existence of solutions is proven.First, the local existence of a weak solution is concluded from existence results for parabolic differential equations with Dirichlet boundary conditions.Additionally, properties of the solution like its non-negativity and the boundedness of one state variable are shown.Due to the inflow term modeling the arriving of T cells from the lymphs, showing a-priori boundedness of the second state variable is the main concern.
The boundedness of the second variable is shown in different steps, starting with proofs of the boundedness of the solution in  1 () and  2 () in Section 3.2.The proofs use different functionals depending on the  1 ()-or  2 ()-norms and they are handling the oppositely acting mechanisms in the reaction function.As a result, we get rough but robust estimates for the  1 ()-and  2 ()-norms of the solution.In the context of liver infections, this result will be interpreted in the light of the total amount of T cells.
The results are used for proving the boundedness of the solution in  ∞ ().Consequently, the global existence of a bounded solution is shown.The boundedness of the solution in  ∞ () is an important property showing how the mathematical proof evokes insight in the application, which is a liver infection, and vice versa the inflammation application feeds back to the mathematics.The immune response in the second state variable, i.e. the amount of T cells, contains the strength of the inflammation.Its upper bound is related to illness and survival of an infected individual.
In Section 4, the quality of used estimates is visualized for different solutions types which are interpreted as different infection courses.The paper finishes with a conclusion of the results and further ideas.

Reaction diffusion infection model with non-local inflow
A model for describing the interaction between virus and T cells during a viral liver infection is presented in Ref. 9 and analyzed in Refs.9-13.The virus population density  = (, ) is named according to the prey in the classical Lotka Volterra model.The cells of the immune system are concluded as T cells.They can be seen as predator for the virus and are therefore named  = (, ).We model the interaction in a part of the liver seen as a bounded domain  ⊂ R  with  = {2, 3}.
According to Ref. 9, the T cells, as the summed cells of the immune system, kill infected liver cells and thus the virus.Both, the T cells and the virus spread out in the liver, modeled by diffusion terms.The virus grow by reproduction in dependency of the local virus amount.The change of the T cell population depends on the total virus load inside the liver, which is modeled by an inflow term  [].
Since the T cells as immune response are produced in the lymphs outside the liver, the T cells arrive in the regarded part of the liver through portal fields, which are bounded sub-domains  ⊂ .Furthermore, the external production of the immune response motivates the dependence of the inflow  = [] on the total amount of virus in the regarded domain , i. e. the inflow  = []() in every point  ∈  depends non-locally on the integral  () = ‖(, ⋅)‖  1 () of  over .

Remark 1 (Modeling Scale).
In the context of liver infections, the area  can be seen as a model for a portal field through which T cells enter a certain part of the liver .The model abstracts from the cell-scale structure of the liver and the involved cells.Nevertheless, we cover some basic structure of a liver by still regarding portal fields in the liver.
We regard, as a simplification, the boundary  of the bounded domain  to be impermeable.This results in zero flux or homogeneous Neumann boundary conditions.
Using as few mechanisms as possible, see Ref. 12, we find the predator-prey model The growth rate  in Ref. 9 describes a logistic growth of the virus with a strong Allee effect, 14 i.e.
The minimal density for the survival of the virus is  min .Otherwise, the virus is locally attacked and it decreases without the secondary immune response from the lymphs.The parameter  is a small parameter fitting the growth in Eq. ( 2) to a pure logistic growth for values  close to 1.
As usual in population dynamics models, the reaction functions in Eq. ( 1) contain terms with opposite monotonicity behavior.The growth term () and the decay term − act oppositely for  in the equation for  , just like the inflow term [] and the decay term −(1 − ) do for  , .
Remark 2. The particular choice of the growth rate makes ( min ) = 0 and (1) = 0, and it is positive between the zeros.Furthermore  behaves asymptotically like 1 −  for large , and we find that  is increasing in the interval [0,  min ].Thus, the minimal value () for  ∈ [0, 1] is (0) = − min ∕.
Opposite to the classical Lotka-Volterra model, the Allee effect allows a population to become extinct.Besides, the Allee effect does not influence qualitatively the system behavior for larger values .Remark 3. Eq. ( 2) norms the capacity of the logistic growth to 1 because () < 0 for all  > 1.There is no loss of generality because the normalization of  is a pure scaling.A possible  with (, ) > 1 at some  decays in finite time below 1. Due to this realistic property of the model, system (1) is suitable only for (, ) ≤ 1.

The non-local inflow term is
is the total amount of virus, and   () is a non-negative function with supp   () =  ⊂  and As a realistic inflow, we consider   to be at least a bounded and piecewise continuous function.A non-smooth example for   is the characteristic function on the subdomain  ⊂  providing 1∕|| for  ∈  and 0 elsewhere.
Analogously, to Eq. (3), we define the integral of the non-negative  over  as This expression gives the total amount of T cells in  and is important for the harm of an infected organism.The influence of  is positive on both populations  and .In contrast, the influence of  on both populations is negative.Additional to the dynamics of the classical predatorprey model, there is a positive influence on  just depending on , compare the thicker line.This might lead to an unbounded growth of , what is part of our discussion.
Remark 4. Since the integral over   () is 1, we see that the total inflow of T cells is proportional to the total amount of virus.
This property of the model reflects the virus-depending strength of the immune response.The proportionality in Eq. ( 6) contains the monotonous increase of strength of the immune system when the total amount of virus increases.
The total amount  () of virus at the time instant  occurs in Eq. ( 3) and results in the non-local inflow term in the reaction diffusion system in Eq. (1).Consequently, the model equations in Eq. ( 1) are only meaningful if the integral in Eq. (3) exists and is finite, i. e. if (, ⋅) ∈  1 ().We show in Section 3.2, that the solutions  and  stay in  1 () after they are once in  1 ().So in particular, we show therewith that no blow-up in  1 () will occur, cf.Section 3.2.These results will imply that both, the total amount of virus and T cells are bounded in the model.
For this investigation, we have a closer look on the mechanisms in model (1).The reaction terms in system (1) contain oppositely acting mechanisms.For , the growth () leads to an increase of  for  ∈ ( min , 1).As an opposite effect, the term − describes a decrease depending on .The equations for  , contains three mechanisms.First,  increases with the total amount of  in the domain .The increase of  is space-depending and takes place in a subdomain  ⊂ .The second mechanism is a decrease −, which depends linearly on .As a third mechanism, the term  corresponds to − in the first equation, compare the classical Lotka Volterra system.
Fig. 1 shows a state chart of the local reaction mechanisms.It is simplified and abstracts from the space dependency of the increase of  by the inflow term  [].
The non-local inflow term [] is a considerate expansion of the classical Lotka Volterra system because the growth of the predator depends directly on the prey in Eq. (1).That enforces the feedback loop in the way, that an increasing predator population slows down its own growth by diminishing the prey population in , compare (−) in Fig. 1.
The interplay of oppositely acting mechanisms leads to interesting solutions.We observe in Ref. 9 that the system (1) has solutions which can be divided into two qualitative different types.On the one hand, there are solutions tending towards zero.On the other hand, we find solutions with a tendency towards a stationary state which is spatially inhomogeneous.The used parameters and the shape and size of the domain  control towards which stationary state the solution is tending.See Refs.9, 10, 12, 13 for further details on the analytical results.
As the model was found in the context of liver infections, we interpret the two qualitative different solution types as different infection courses.Solutions with a tendency towards zero are associated with healing courses, see Fig. 2. The immune system is able to kill all infected cells during an active phase and therefore, the virus vanishes.
Afterwards, the immune reaction fades out and the T cell amount tends towards zero as well, see Fig. 2(b).
Solutions with tendency towards stationary spatially inhomogeneous states are interpreted as persisting or chronic infections, compare Fig. 3.After an active phase with a strong immune reaction, the T cell amount decays, but does not vanish and the virus persists in the liver.In the stationary phase, there is still virus in the whole domain , see Fig. 3(a), and T cells as well, see Fig. 3(b).
In addition to Figs. 2 and 3, where space-dependent solutions for a fixed time are displayed, Fig. 4 shows the trajectories of the total virus  () and T cell populations  () of different infection courses over the time.
Fig. 4 shows, that the total populations tend towards a stationary state in both cases.Together with the space dependent Fig. 3, Fig. 4 shows the tendency of the solution towards a spatially inhomogeneous stationary distribution for a chronic infection course.This is in accordance with pathological images, compare, 15 where the spread of T cells vary in different parts of the liver tissue.

Existence
The model in Eq. ( 1) reflects biological structures, see Remark 1, and uses a non-local and space-dependent term for modeling the biological structure of the application.The analysis of this model leads to an interesting new problem which cannot be handled easily by standard approaches.Besides, we are interested in a proof using interpretable intermediate steps for gaining a deeper understanding of the systems dynamics.
Of course, there are many theorems for the existence of a solution of a reaction diffusion system or more general a parabolic partial differential equation.In this section, we mention some important results on the existence of solutions for reaction diffusion equations and explain, why they cannot be applied directly to the system (1).
There are at least two main approaches often used in proofs of existence theorems for parabolic partial differential equations.One approach uses fixed point theorems, like the Banach fixed point theorem, the Brouwer fixed point theorem and from this following the Schauder and the Leray-Schauder fixed point theorems. 16The second approach uses semigroup theory, see Refs.17, 18.
The first approach using fixed point theorems can be found for example in Ref. 16, p. 536.There, the existence and uniqueness of solutions is shown under the requirement, that the local reaction function  is Lipschitz continuous with respect to  = (, ) T .This requirement is used for showing the contraction of the operator for the fixed point theorem.Additionally, the theorem in Ref. 16 requires Dirichlet boundary conditions.
In Ref. 19, p. 188, an existence theorem for a reaction diffusion system with Lotka Volterra reaction terms is shown.The proof is based on the Schauder fixed point theorem and uses a-priori bounds for the state variables.
There are several proofs for monotonous reaction functions as well, see Ref. 20, p. 120.
Unfortunately, the reaction functions in Eq. ( 1) are neither globally Lipschitz continuous with respect to  and , nor monotonous.Even if  is bounded by construction by an upper limit 1, an a-priori upper bound for  is not obvious.We show the existence of a global upper bound in Section 3.3.
Existence results using a semigroup approach are based on limited growth conditions, for example 18, p. 276 or 21, p. 75 .Due to the nonlocal integral term, the nonlinear terms and the unavailable a-priori bound for , the system in Eq. ( 1) does not fulfill the requirements for these existence results.As already mentioned, the existence of a finite a-priori bound for  and therefore the boundedness of  in the  ∞ -norm is a relevant question concerning the application in modeling liver infections.   2 but  = 0.7 and  = 0.9.The time steps are  0 = 0 (bright),  1 = 10.5,  2 = 30 (dark).Starting with the same initial conditions as in Fig. 2, the virus and the T cells persist in the whole domain.The T cell amount is higher around the portal field .There is nearly no difference between the spread at  1 and at  2 .Results for reaction diffusion systems with non-local effects can be divided into results for nonlinear diffusion and nonlinear reaction terms.The global existence of solutions for systems with nonlinear diffusion, with homogeneous Dirichlet boundary conditions is shown in Ref. 22.The results yield if the solutions are non-negative and the total mass is controlled.Additionally, an a-priori estimate in the  1 ()-norm for the nonlinear reaction functions is required.
In Ref. 23, the reaction diffusion equation with a non-local term and with homogeneous Dirichlet boundary conditions is analyzed.The global existence of non-negative solutions is shown for any  > 1.
As a third example, the global existence of solutions of the general formulation where  is a parabolic operator and  is bounded in the  2 ()-norm is shown in Ref. 24.
The results are mainly for single equations instead of systems, and the requirements are not fulfilled for system (1).Again, the system with coupled equations and an integral term require new approaches for proving the existence of globally bounded solutions.
The adaption of the named existence theorems on our system (1) requires -if possible at all -severe modifications on a technical mathematical level.However, by proving the longterm existence, we aim to develop a deeper understanding of the infection application.Therefore, we present a step by step proof and accompany it by biological and medical applications.Now, we show the existence of solutions and their boundedness in  ∞ (), which allows a point-wise estimation of the maximal virus and T cell amounts.The section has the following structure.First, the existence of a weak solution for a small time span [0,  ) is shown.We discuss some basic properties of such solutions like non-negativity of  and  and boundedness of .These properties are important for modeling purposes as negative values are not interpretable in the context of densities of virus and T cells.
In Section 3.2, the boundedness of  in  1 () is shown.This result shows a boundedness of the total amount of virus and T cells at a certain time.Afterwards and building up on this result, the boundedness of the norm ‖‖  2 () is proven.
Finally, in Section 3.3 the boundedness of the norm ‖‖  ∞ () is shown and using it, the global longterm existence of weak solutions of Eq. ( 1) is shown.
In Ref. )), which is bounded in a suitable chosen time interval  ∈ [0,  ).Define () =  (()) as a right-hand side for the general parabolic system Due to the boundedness of  in the limited time interval and the smoothness of  , the function  is regular in the sense, that  ∈  2 ([0,  );  2 (, R 2 )).
The weak solution (, ) of Eq. ( 1) fulfills some basic properties.Proof.Again, we regard the maximum max ∈ (, ) = (,  max ).If it is larger than 1, the logistic growth () is strictly negative at the point  max .Since  has its maximum at  max , the diffusion term fulfills  ≤ 0. At the same time,  is increasing, so that the predator term − is larger than 0, and the maximum max ∈ (, ) passes the value 1 with a non-zero time derivative at a finite time instant  1 .□ The proof shows that  does not tend to 1, but rather passes 1.That means that the virus decays under its capacity, whenever an active immune response exists.We formulate this observation in a next corollary saying that  becomes smaller than 1 together with a non-vanishing  on some sub-domain of .Since the inflow [] is positive in both cases,  increases, and the predator term − is strictly negative in  for all  >  1 .Therefore the assertions are fulfilled for every  2 >  1 with sufficiently small  2 − 1 .□

Corollary 10. All bounded and non-vanishing initial values allow to find a time instant 𝑡
In the following, we assume initial conditions (0, ) ≤ 1 for all  ∈ .As shown in Corollaries 9 and 10 and according to the formulation of system (1), this is not a restriction.
We interpret Corollary 10 in the light of application.Even if there would be a higher amount of virus than the upper limit allows, the additional virus vanish by a negative growth term and spread out by diffusion.The negative growth can be interpreted as a decay due to a limited number of free liver cells where the virus can attach.
Proof.Due to Lemma 8, the solution (, ) is bounded by 1. Integration of both sides of  ≤ 1 gives  () ≤ ||.□ With these results, we found a (weak) solution for a time interval [0,  ), which is non-negative and at least one component of the solution, namely , is bounded.The increase of the second component  depends on the  1 ()-norm  of .Hence, until now,  could still grow over all bounds.
Consequently, we have to show that the increase of  happens simultaneously to a decrease of  , cf.Fig. 1, and that this simultaneity makes  to be bounded in the different norms.
Since we will need it in the next section for showing that blow-ups of the solution of system (1) do not occur, we prove that  is not only bounded by 1 but it is sufficiently remote from 1 after some time.The medical background suggests that a virus density close to 1 provokes an increase of the immune response.Hence, the virus density decreases.This slows down the influx of T cells again, compare the opposite directions of the mechanisms in Fig. 1.The following Lemma 12 will give a very rough estimate for this observation.
But first, we consider the solution  aux =  aux () of the auxiliary stationary problem for  ∈ , The function   () ≥ 0 is at least piecewise continuous and not vanishing in the whole domain .Consequently  aux is continuous, bounded and positive.Since   () is positive only in the influx region , there is some value  thr > 0 with  aux () ≥  thr for all  ∈ .Therewith, we are prepared to prove the announced Lemma.
Lemma 12. Let ,  be weak solutions of (1).For all  ≥ 0, there is a  with 0 <  < 1 and a time  3 with for all  ≥  3 .
Proof.First, we show that there is at least one  3 for which Eq. ( 9) is fulfilled.
Assume, there would be no such  3 .Then,  must be equal 1 almost everywhere in supp  ⊂  for all time .As a solution of Eq. ( 1),  is continuous with respect to .Consequently,  must be equal 1 in  and we get the rough estimate  () ≥ ||.Now, the evolution of  in Eq. ( 1) reads and after a transient phase, we get (, ) ≥  () aux () ≥ || aux () and thus (, ) ≥ || thr for all  ∈ .(10)   Finally, the first equation in system (1) reads So, Eq. ( 10) implies  , < 0 for all  ∈ , what contradicts the assumption  = 1 in .Consequently, there is at least one time instant  3 fulfilling Eq. ( 9).
If we now assume that  grows again after  3 so that the estimate ( 9) is hurt for every  < 1 at some  4 , that would mean  gets arbitrarily close to 1 in .This is again a contradiction to () −  () aux () ≤ () − || aux () < 0 at this time instant  4 .□ In the next steps, we show, that there exists an upper bound for  as well.First, we show, that  is bounded in  1 () for  ∈ (0,  ).Next, we expand this property for all times  > 0. As an intermediate step, we show (, ⋅) ∈  2 ().Finally, by using the stationary solution of another related elliptic equation for a stationary problem, we prove that  is bounded and smooth for all times , (, ⋅) ∈  ∞ ().
In this section, we show, that  is not growing to infinity for  ∈ [0,  ).
This theorem says that the  1 ()-norm of a solution (, ) of system (1) stays in a bounded region, namely within the trapezoid , compare Fig. 5, as long as a weak solution exists.
Proof.The time derivative of the functional  =  +  is with system (1)  (11).In particular,  cannot pass  up when it is once lower than  up with  ≤ ||.Consequently, the  1 ()-norm ( ,  ) of a solution (, ) stays in  when it starts in .
If now then () ≤  up for all admissible .If otherwise (0) ≥  up , we have shown that  decreases until () is smaller than  up .Finally, () ≤ max{(0),  up } =  up for all admissible  and all initial values allow to construct a suitable  where the solution stays in.□ In Fig. 5, the trapezoid  is shown in the phase space of ( ,  ).The arrows show the direction of the dynamics given by the reaction term in system (1).The arrows of the dynamics point inside  or at least not to the exterior, especially at the upper bound of  .Remark 14.Since || > 0, the  1 ()-norm  () of  is bounded by for all  ∈ [0,  ).
That means that the summed strength of the immune response is bounded in a bounded time interval.
The  1 ()-norms  ( −) and  ( −) are for any  > 0 inside of . depends only on the initial values, but it is independent of the time  and the solutions  and  theirselves.Consequently, the  1 ()-norms  ( ) and  ( ) are inside of  as well.The values ( , ) and ( , ) can be seen as new initial data of system (1).By induction, the  1 ()norms  () and  () are inside of  for every  > 0 and the  1 ()-norm  () of (, ) is bounded by  up in Eq. ( 12) for all time  > 0.
This results shows the boundedness of the total amount of T cells at any time, not only for a limited interval.
The amount of T cells  is not only bounded in the sense of  1 () but also in the sense of  2 ().This can be shown by regarding the time derivative of the functional Theorem 16.Let (, ) be a solution of system (1).Then, the  2 ()-norm of  is bounded for all  > 0. Proof.The time derivative of the functional  is Using Green's first identity and the zero-flux boundary conditions, we get Further, Remark 4 provides an estimate for the integral of [], which is with  max = max ∈   () according to Eq. ( 4).Now, Remark 14 assures with the constant  =  max || up and the weighted mean value () defined by The mean value () fulfills 0 < 1 −  ≤ () because of Lemma 12. Finally, the functional  obeys the linear differential inequality with a positive decay rate 2() which stays remote from 0. Eq. ( 14) is a first order differential inequality, compare, 25 and  () is bounded by the solution of the linear first order differential equation Thus, the largest possible accumulation point of  is  2(1−) , and the functional  is bounded by  (1−) after a transient phase.Later in Section 4, we will use the estimate for showing numerically the precision of the estimates.□ Theorems 13 and 16 show, that the  1 ()-and the  2 ()-norms of  are not only bounded for a time interval [0,  ) but for all time  > 0. So in these norms, the solution is not blowing up.

𝐿 ∞ (𝛺) Bounds and global existence
In this section, we show the boundedness of  in  ∞ () for all  > 0. With the boundedness of (, ), the existence of a solution (, ) T with finite values is shown for all  > 0.
We will prove, that there exists a value  max with (, ) ≤  max for all  ∈  and all  ∈ [0, ∞).For this purpose, a stationary problem is defined.Let  ⋆ =  ⋆ () be a solution of System ( 16) fulfills the solvability condition because the forces are equalized, see Eq. ( 4) and Remark 17.Since the right-hand side   () − 1 || in Eq. ( 16) is a bounded piecewise continuous function and thus in  2 () ⊂  −1 (), the existence of a weak solution  ⋆ ∈  1 () is ensured, compare. 26mark 18.The solution  ⋆ of Eq. ( 16) has a free additive constant as always in pure Neumann problems.In the following, we fix just one  ⋆ with ‖ ⋆ ()‖  1 () = 0. Now, we will show that the population  = (, ) in Eq. ( 1) does not grow to infinity.Even having already estimates for its  1 ()-norm, cf.Theorem 13, and for its  2 ()-norm, cf.Theorem 16, it is not trivial to give a pointwise bound.Before we will do that in the later Theorem 23, we collect some auxiliary results about solutions of partial differential equations with homogeneous Neumann boundary conditions.The mathematical argumentation starts with the Green's function  = (, , , ) of Eq. ( 17), which is dominated by the singularity of the standard heat equation.Due to the Neumann boundary condition, there are no additional source terms at the boundary.The Laplacian () is the convolution of    with the bounded function  .This convolution can be estimated by a sum of spatial integrals over small domains and afterwards by time integration leading to terms in the Gauss' error function.Due to its technical effort, we omit the argumentation of the physically proven assertion of Lemma 19.By the way, another possible argumentation uses a discretization of the Eq. ( 17), where the eigenvalues and eigenvectors of the discretized differential operator −( + ) can be estimated in a similarly technical argumentation.Then the limit case of a temporal step size tending to zero provides the assertion of Lemma 19 for every spatial discretization, and since  is bounded also the limit situation of a vanishing grid size.with the smooth and bounded potential Φ for a constant source term.
There is a value  3,max < ∞ so that the condition ‖‖  1 () ≤  2 is not fulfilled for any  3 ≥  3,max .Consequently, we get This lemma shows the boundedness of a stationary problem which displays in parts the inflow of T cells in a certain region.In the next step, we use this result for showing the existence of an upper bound for  in a time-dependent setting which still abstracts from the coupled reaction diffusion system in (1).
In this section, we have proven the boundedness of the solution of Eq. ( 1).While the boundedness of  was a result of the used growth function and therefore allows interpretation as a concentration, the boundedness of  was not obvious.Using the oppositely acting mechanisms in the reaction functions and the boundedness of , we first showed the boundedness of  in  1 ().
We provided a bounded estimate for the  2 ()-norm of  by using the mean-value theorem of integration and the boundedness of ‖(⋅, )‖  1 () .
For proving the boundedness of  in  ∞ (), we separated (, ) =  ⋆ () () + ṽ(, ) into different functions.One component,  ⋆ , of the functions was the solution of a stationary problem covering the spacedependent function modeling the inflow area of the liver structure.By showing the boundedness of all components of , we proved in Theorem 23 that  has a finite maximal value.
Applied to the modeling of liver infections, the result of Remark 24 says that the amount of T cells is bounded by a finite value.The T cells attack the virus by triggering the programmed cell death.This leads to inflammation in the liver tissue and can cause secondary diseases like cancer.Besides, a too high amount of T cells might cause a sepsis.Remark 24 gives the fact, that the immune reaction is bounded but does not give a finite value.Therefore, Remark 24 justifies the use of model in the sense, that the immune reaction remains bounded for all time.It is a first step towards quantitative and finer estimates for  max which contain information about the occurrence of sepsis.
Therefore, the next section provides numerical evaluations on the  1 () and  2 () estimates.

Numerical evaluation of the estimates
Oftentimes, estimates used in analytical results are rather rough.In this section, we show numerical simulations of the estimates and the exact value.The simulations are based on the simulations in Figs. 2  and 3.Those use a semi-discretization by finite differences for the space and Runge-Kutta based methods for the time.As the finite differences discretization of the Laplacian might lead to a stiff ordinary differential equation problem, a standard solver for stiff ordinary differential equations is used in Matlab.The numerical evaluation of the estimates uses finite sums instead of the integrals in accordance with the space and time resolution used for solving the partial differential equations.
First, we evaluate the estimation of the domain  as maximal  1 ()norms.In Fig. 6, the trajectories of the two solutions from Figs. 2 and 3 provide the total amount  and  .They are compared to the estimated  following Theorem 13.Fig. 6 shows, that the upper bound of  +  ≤  up is a rather rough estimate for the  1 ()-norms of the solutions.In the numerical simulations in Figs. 2 and 3, which are as well used in Fig. 6, the initial conditions are (0, ) ≡ 1 and (0, ) ≡ 0. A solution with larger initial conditions  (0) would reach closer to the upper bound given by  =  up .As shown in Theorem 13, the  1 ()-norms of every solution with ( (0),  (0)) ∈  stay in .
In the cases of the two regarded simulations in Figs. 2 and 3 of Eq. ( 1), the initial value  (0) is zero, because (0, ) = 0 for all  ∈ .Therefore, we compare the  () to the functional In both cases in Fig. 7, the estimation  is rather large compared to the functional  ().Nevertheless, the estimate is a good approximation of scale of the maximal value of  .
The numerical simulations show that the used estimates are rather loose even if they were sufficient for gaining the analytical existence results.The estimations exceed the values of the numerical solutions.In context of liver infections, the estimations can be regarded as worst case scenario.A medical treatment in the light of the worst case scenario might lead to a longer lasting infection course but decrease the risk of a sepsis.

Conclusions
With the aim to modeling the dynamics of liver infections as an interplay between virus and T cells, a reaction diffusion system was presented in Ref. 9. A non-local term in the reaction function describes the inflow of T cells depending on the total virus amount in the domain.The model abstracts from the cell scale with many unknown mechanisms to a mesoscopic length scale.On this scale, the mathematical description contains a space-dependent term which leads to a new problem concerning the analysis of reaction diffusion equations.Additionally, the reaction terms contain oppositely acting mechanisms resulting in a feedback loop for the increase of  cells.The longtime behavior of the solutions of the model depends on the used parameters.On the one hand, there are solutions with a strong tendency towards zero.These solutions are interpreted as healing infection courses.On the other hand, there are solutions with a tendency towards a spatial inhomogeneous steady state.These solutions are interpreted as chronic infection courses with some virus persisting in the liver.One quantity of the model gives the amount of T cells which trigger the cell death of infected liver cells.This effect leads to further mechanisms known as inflammation of the tissue.Inflammation might be critical as the danger of sepsis is connected to strong inflammation.Therefore, it is important to know about upper bounds for the modeled quantities.
The aim of this paper was to prove the existence of bounded solutions for all time and to provide a proof with interpretable intermediate steps.First, some results on the existence theory for parabolic partial differential equations are summarized.As the focus lies on the  interpretation of the gained results are therefore, the steps for proving longtime existence and boundedness of solutions are explained.
We started with a local existence theorem and some properties of a weak solution.Then, we showed the boundedness of the solution in the  1 ()-norm and in the  2 ()-norm.Both results are based on the interplay of the two species in the population dynamics model and the oppositely acting mechanisms of growth and decay.While the local boundedness of  was a result of scaling and a limited growth, the dynamics of the second state variable are more complex and depending locally on the integral of .For proving boundedness of the second variable, the pure boundedness of the first variable is not enough: We need working predator-prey mechanisms leading to a decay of the first variable away from the maximal capacity.Additionally, the non-negativity of both state variable is shown.This is important for interpreting the state variables biologically.The  1 -bound for  gives a maximal total amount of T cells in the regarded domain and therefore information on the maximal inflammation.We then defined a stationary problem for showing the boundedness of the solution in the  ∞ ()-norm.This approach was based on a decomposition of the solution for the T cells into a stationary and a time-dependent problem.
In Section 4, we evaluated the sharpness of the used estimates in the proofs.The numerical simulations show that the estimates are rather loose for the regarded cases.Nevertheless, the estimations provide insight: In the light of the application, modeling liver infections, the estimation can be seen as a worst case scenario.The boundedness of the  ∞ ()-norm of  is a feature uprating the model for liver infections.
A further investigation could be the improving of the used estimated such that the difference between estimate and real value of the functionals becomes smaller.Another possible extension is the application of the estimates for a wider class of integro-partial differential equations.The gained estimations can be used for planning treatment more effectively and with less harm for the patients.First results on modeling treatment plans by hands of Eq. ( 1) can be found in Ref. 27.

Financial disclosure
None reported.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
) with a growth function () describing the non-linear growth of the virus in absence of other mechanisms and the non-local inflow  = []() of T cells.The constants  and  describe the strength of diffusion.The reaction diffusion system in Eq. (1) contains the predator term  diminishing the virus in presence of the immune response , and the decay term (1 − ) describing the fade out of the immune response in absence of any virus.

Fig. 1 .
Fig. 1.State chart for the reaction mechanisms of system (1) for realistic  ∈ ( min , 1).The influence of  is positive on both populations  and .In contrast, the influence of  on both populations is negative.Additional to the dynamics of the classical predatorprey model, there is a positive influence on  just depending on , compare the thicker line.This might lead to an unbounded growth of , what is part of our discussion.

Fig. 2 .
Fig. 2. Numerical simulation with a solution interpreted as healing infection course.Used parameter values are  min = 0.05,  = 0.01,  = 0.9,  = 3.7,  = 0.2,  = 0.6 and  = 0.3.The initial conditions for  0 = 0 (bright mesh) show the amount of virus and  cells right after the activation of the immune response.In (b) T cells enter the domain through an area  around ( 1 ,  2 ) = (1, 1).The virus is killed by the T cells and decays for  1 = 0.75 and  2 = 3 (dark mesh).The amount of T cells reduces due to the very low virus concentration.Both population vanish after an active phase.

Fig. 3 .
Fig.3.Numerical simulation with a solution interpreted as chronic or persisting infection course.Used parameter values are the same as in Fig.2but  = 0.7 and  = 0.9.The time steps are  0 = 0 (bright),  1 = 10.5,  2 = 30 (dark).Starting with the same initial conditions as in Fig.2, the virus and the T cells persist in the whole domain.The T cell amount is higher around the portal field .There is nearly no difference between the spread at  1 and at  2 .

Fig. 4 .
Fig. 4. Numerical simulations according to those in Figs. 2 and 3. (a) Total virus  and T cell amount  during a chronic or healing infection course over the time.(b) Summed dynamics of a healing or chronic infection course in phase space.

𝒙∈𝛺Theorem 23 .
(, ⋅) grows at most linearly.So the following proof excludes an infinite growth of  for  → ∞.Proof.System(19) is a linear differential equation and the solution  decomposes into  =  hom +  part .The function  hom obeys the homogeneous equation with  ≡ 0 and fulfills the initial conditions  0 .The function  part solves the system (17) from Lemma 19.The function  hom follows the maximum principle max ∈,≥0 | hom (, )| = max ∈ | 0 ()| and stays bounded.Lemma 19 says that  part (, ⋅) has a bounded Laplacian | part (, )| ≤  1 for all  ∈ .Lemma 21 assures that  part is bounded for all times by a  max ∈ R. Together with the boundedness of  hom , we find |(, )| ≤ max ∈ | 0 ()| +  max .□ Following, we adapt this result for the coupled reaction diffusion system modeling the dynamics of a liver infection.The solution  of Eq. (1) is bounded by a finite value  max .Proof.We decompose  = (, ) into (, ) =  ⋆ () () + ṽ(, ) and the evolution equation for  in Eq. (1) transforms into

Fig. 6 .
Fig. 6.Comparison of the trajectories of different solutions in phase space ( ,  ) and the trapezoid .(a) Healing course, see Fig. 2 for the parameters.The upper value  up is given by  up = 19.833.(b) Chronic course, see Fig. 3 for the parameters.The upper value  up is given by  up = 4.
The proof follows Ref. 16, Theorem 9.2.2, p. 536 and Ref. 16, Theorem 3, p. 378 with the mentioned adaptions of the boundary conditions.