Abstract
The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in Viguerie et al. (Appl Math Lett 111:106617, 2021); Viguerie et al. (Comput Mech 66(5):1131–1152, 2020), where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in Auricchio et al. (Math Methods Appl Sci 46:12529–12548, 2023) where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed \(L^p\)-energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.
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1 Introduction
1.1 Mathematical analysis of quasilinear SEIRD models
Modelling the spatial spread of infectious disease is a classical topic and has recently attracted much more interest from the mathematical community due to the Covid-19 epidemic. Classical models include the susceptible-infected-removed (SIR) system and its variants, such as susceptible-exposed-infected-removed (SEIR) or susceptible-exposed-infected-removed-deceased (SEIRD). To account for spatial heterogeneity, a compartmental model was proposed in [31, 32] where the diffusion coefficients are highly heterogeneous and depend on the total population density, leading to a quasi-linear reaction-diffusion system. The simulations therein showed a strong qualitative agreement from the forecast using the model and collected data in Lombardy.
Let \(\Omega \subset {\mathbb {R}}^N\), \(N\ge 1\), be a bounded domainFootnote 1 with smooth boundary \(\partial \Omega \) such that \(\Omega \) lies locally on one side of \(\partial \Omega \). Let s(x, t), e(x, t), i(x, t), r(x, t), and d(x, t) denote the susceptible, exposed, infected, recovered, and deceased population densities, respectively, at spatial position \(x\in \Omega \) and at time \(t>0\), and let \(n(x,t):=s(x,t)+e(x,t)+i(x,t)+r(x,t)\) be the total living population density. The proposed model in [31] reads as
where the positive diffusion coefficients \(\nu _s, \nu _e, \nu _i, \nu _r\), the birth rate \(\alpha \), the inverse of the incubation period \(\sigma \), the asymptomatic recovery rate \(\phi _e\), the infected recovery rate \(\phi _r\), the infected mortality rate \(\phi _d\), the asymptomatic contact rate \(\beta _e\), the symptomatic contact rate \(\beta _i\), and the general mortality rate \(\mu \) are positive and may depend on time and space.
The mathematical analysis of (1.1) was first studied in [1]. One distinguishing characteristic of this system, in comparison with many other variants of SIR models in the literature, is that the diffusion of all species depends on the total population density n, which makes (1.1) a quasilinear reaction-diffusion system. This also brings possible degeneracy to the diffusion operators and makes the problem more challenging. The global existence and boundedness of solutions to (1.1) was shown in [1] under the following assumptions and modifications:
-
(i)
all diffusion rates are the same,Footnote 2 i.e. \(\nu _s = \nu _e = \nu _i = \nu _r = \nu \),
-
(ii)
the terms \((1-A_0/n)\) and \(\phi _di\) are replaced by a non-singular function A(n), for instance \(A(n) = (1-A_0/n)_+\), and \(\phi _dni\), respectively.
The replacement of \((1-A_0/n)\) avoids the singularity when n gets close to zero. A closer examination later reveals that this is not necessary since one can estimate \((si)/n \le s\) and \((se)/n\le e\). The assumptions of the same diffusion rate and replacement of \(\phi _di\) are, on the other hand, essential as they help to obtain a key property that the total population density n is bounded pointwise from below by a constant for all time, provided the initial density \(n_0\) is also bounded from below. Indeed, summing the equations of s, e, i, and r, keeping in mind the aforementioned modifications, yields an equation for n of the form
from which the lower bound for n on finite time intervals follows from the lower bound of initial data. With this lower bound of n, the diffusion operators become non-degenerate and the analysis can be carried out in a standard way. This strategy used in [1] seems to break down when the diffusion rates are different, but it still applies when the term \(-\phi _d i\) stays in place, since in this case
So n is bounded from below on finite time intervals if \({n(x,0)\ge \varrho >0}\) for all \(x\in \Omega \).
In this paper, we set out to remove the assumptions (i) and (ii) above and show the global existence and boundedness of solutions to the original system (1.1). In fact, we show the results for a much larger class of quasilinear reaction-diffusion systems which contains (1.1) as a special case. Our key idea is to exploit a recently developed \(L^p\)-energy approach in e.g. [12, 23] which does not require any lower bound on n except for its natural nonnegativity. In the next subsection, we provide the general setting while the main results and ideas are presented in Sect. 1.3.
1.2 Problem setting
Let \(1\le N\in \mathbb {N}\), and \(\Omega \subset \mathbb {R}^N\) be a bounded domain with Lipschitz boundary \(\partial \Omega \). Let \(2\le m\in \mathbb {N}\). In this paper, we study the global existence and boundedness of the following quasi-linear reaction-diffusion system of concentrations \(u=(u_1, \ldots , u_m)\), for any \(i\in \{1,\ldots , m\}\),
where \(\eta \) is the unit outward normal vector on \(\partial \Omega \), initial date \(u_{i,0}\) are bounded and non-negative, the diffusion matrix \(D_i\): \(\Omega \times [0,\infty )\rightarrow \mathbb {R}^{N\times N}\) satisfies
for some \(\lambda >0\), and for each \(T>0\),
and \(\Phi (u)\) satisfies:
-
(Q1)
\({\Phi : \mathbb {R}^m_+\rightarrow \mathbb {R}_+}\) is continuous;
-
(Q2)
There is some \(b\ge 0\) and \(M>0\), such that
$$\begin{aligned} \Phi (u)\ge M \,u_i^b, \quad \forall u \in {\mathbb {R}}_+^m \text { and } i=1,\ldots ,m; \end{aligned}$$ -
(Q3)
There exist \(\pi >0\) and \({\widetilde{M}}>0\) such that
$$\begin{aligned} \Phi (u) \le {\widetilde{M}} \left( 1+\sum _{i=1}^{m}u_i^{\pi }\right) , \quad \forall u\in {\mathbb {R}}_+^m. \end{aligned}$$
The nonlinearities \(f_i(x, t, u): \Omega \times \mathbb {R}_+\times \mathbb {R}^m_+\rightarrow \mathbb {R}\) satisfy the following conditions:
-
(A1)
For any \(i=1,\ldots ,m\) and any \((x,t)\in \Omega \times \mathbb {R}_+\), \(f_i(x,t,\cdot ): \mathbb {R}^m\rightarrow \mathbb {R}\) is locally Lipschitz continuous uniformly in \((x,t)\in \Omega \times (0,T)\) for any \(T>0\);
-
(A2)
For any \(i=1,\ldots ,m\) and any \((x,t)\in \Omega \times \mathbb {R}_+\), \(f_i(x,t,\cdot )\) is quasi-positive, i.e., \(f_i(x,t,u)\ge 0\) for all \(u\in \mathbb {R}^m_+\) with \(u_i=0\) for all \(i=1,\ldots ,m\);
-
(A3)
There exists \(c_1,\ldots ,c_m>0\) and \(K_1,K_2\in \mathbb {R}\) such that
$$\begin{aligned} \sum _{i=1}^{m}c_if_i(x,t,u)\le K_1\sum _{i=1}^{m}u_i+K_2, \quad \forall (x,t,u)\in \Omega \times \mathbb {R}_+\times \mathbb {R}^m_+; \end{aligned}$$ -
(A4)
There exist \(K_3>0, r>0\), and a lower triangular matrix \(A=(a_{ij})\) with positive diagonal entries, and nonnegative entries otherwise, such that, for any \(i=1,\ldots ,m\)
$$\begin{aligned} \sum ^i_{j=1}a_{ij}f_j(x,t,u)\le K_3 \left( 1+\sum _{i=1}^{m}u_i^r\right) , \quad \forall (x,t,u)\in \Omega \times \mathbb {R}_+\times \mathbb {R}^m_+ \end{aligned}$$(we call this assumption an intermediate sum of order r);
-
(A5)
The nonlinearities are bounded by a polynomial, i.e., there exists \(l>0\) and \(K_4>0\) such that \(\forall i=1,\ldots , m\),
$$\begin{aligned} |f_i(x,t,u)|\le K_4 \left( 1+\sum _{i=1}^{m}u_i^l\right) , \quad \forall (x,t,u)\in \Omega \times \mathbb {R}_+\times \mathbb {R}^m_+. \end{aligned}$$
The local Lipschitz continuity (A1) of the nonlinearities implies the existence of a local solution to (1.2) on a maximal interval \([0, T_{\max })\). The quasi-positivity assumption (A2) assures that the solution to (1.2) is non-negative (as long as it exists) if the initial data is non-negative, which is a natural assumption since we consider here \(u_i\) as concentrations or densities. The assumption (A3) gives an upper bound on the total mass of the system. Reaction-diffusion systems satisfying (A1), (A2) and (A3) appear naturally in modeling many real life phenomena, ranging from chemistry, biology, ecology, or social sciences. Remarkably, these natural assumptions are not enough to ensure global existence of bounded solutions as it was pointed out by counterexamples in [26] and [27] even for semilinear systems, i.e. \(\Phi (u)\equiv 1\) and \(D_i(x,t)\equiv D_i\) in (1.2) where \(D_i\ne D_j\) for some \(i\ne j\). The study of global existence of semilinear systems, i.e. (1.2) with \(\Phi (u)\equiv 1\), has made considerable progress in the last decade, see e.g. [4, 8, 10,11,12, 15, 17] and the survey [25].
One can readily check that all assumptions (Q1)–(Q3), (A1)–(A5) are fulfilled for the SEIR system (1.1) (excluding the equation of d as it is uncoupled) provided that the diffusion rates are bounded from below by positive constants, and all other rates are nonnegative and bounded functions of x and t. Indeed, by writing \(u_1 = s\), \(u_2 = e\), \(u_3 = i\), \(u_4 = r\), and \(u = (u_1,\ldots ,u_4)\), we have \(\Phi (u) = u_1+u_2+u_3 + u_4\) satisfying (Q1)–(Q3) with \(b = \pi = 1\). Now (A1)–(A3) and (A5) are obviously fulfilled. To check (A4), we observe the first nonlinearity
where we used \(si/n \le s\) and \(se/n\le e\) and the nonnegativity of the rates. Thanks to the boundedness of the rates, we see that the first nonlinearity is bounded above by a linear combination of all components. It’s immediate that the third and forth nonlinearities, as well as the sum of the first and second nonlinearities are bounded by a linear combination of all components, since all nonlinear terms are cancelled out when summing. Hence, by choosing \(r = 1\) and the matrix
we see that (A4) is satisfied. It is noted that all these assumptions also hold if we replace \(-\phi _di\) in (1.1) by \(-\phi _d in\) as done in [1]. Therefore, (1.1), as well as the modified one in [1], is indeed a special case of the general system (1.2).
1.3 Main results
Let us start with the first main result about the global existence and boundedness of system (1.2).
Theorem 1.1
Assume (1.3), (1.4), (Q1), (Q2), (Q3), (A1), (A2), (A3), (A5) and (A4) with
Then for any nonnegative, bounded initial datum \(u_0\in (L^{\infty }(\Omega ))^m\), there exists a global weak solution to (1.2) with \(u_i\in L^{\infty }_{loc}(0,\infty ; L^{\infty }(\Omega ))\) for all \(i=1,\ldots ,m.\) In particular, if \(K_1<0\) or \(K_1=K_2=0\), then the solution is bounded uniformly in time, i.e.
Remark 1.2
-
When \(b=0\), we have \(\Phi (u)\ge M\). By letting \(\Phi (u)=1\), our results cover the result in [12, Theorem 1.1].
-
It is noted that under the assumption \(K_1<0\) or \(K_1 = K_2 = 0\), we can get the \(L^1(\Omega )\)-norm of the solution bounded uniformly in time (see Lemma 3.1). This fact can be used to show the uniform-in-time bound (1.6). As a result, if we can use other ideas or use other structure of the system to get the \(L^1(\Omega )\)-norm of the solution bound uniformly in time, we can remove the assumption \(K_1<0\) or \(K_1 = K_2 = 0\).
-
It is noted that the key idea to prove global existence is to construct the \(L^p\)-energy function. Actually, the intermediate sum condition (A4) is essential to construct the \(L^p\)-energy function.
The proof of Theorem 1.1 is based on an \(L^p\)-energy approach. The traditional idea has been to construct an energy function that is decreasing or at least bounded in time for the (1.2) of the form
It is noted that if \(h_i(z) \sim z^p\), this yields an \(L^p\)-estimate of the solution, and for p large enough, using a bootstrapping procedure, one obtains an \(L^\infty \)-estimate which implies global existence. Unfortunately, this approach is very likely to fail under the general assumptions (A3)–(A4), except for some very special cases. The duality method (see e.g. [4, 13, 19, 20, 22, 25]) is very efficient when dealing with systems with constant or smooth diffusion coefficients. Using this method, one gets an \(L^{2+\varepsilon }(\Omega \times (0,T))\)-estimate from the mass control condition (A3). Combining this initial estimate, another duality argument (see [22]) and bootstrap argument, using the intermediate sum condition (A4) and the growth assumption (A5), one can obtain an \(L^\infty (\Omega \times (0, T))\)-estimate that ensures global existence (more details can be found in [22]). It is noted that this method does not extend to the case of merely bounded measurable or quasi-linear diffusion coefficients, unless some additional regularity assumptions are given (see [2, 6]). Recently, an \(L^p\)-energy approach was given in [12, 23], whose preliminary ideas have been used previously in, e.g., [14, 24]. It is remarked that a similar method has also been successfully applied to cross-diffusion systems, see [16]. The core idea of this \(L^p\)-energy method is to construct a generalized \(L^p\)-energy function of the following form:
where \(p\in \mathbb {N}\) with \(p\ge 2\) and
with
where \(\theta =\left( \theta _1, \ldots , \theta _m\right) \) and \(\theta _1, \ldots , \theta _m\) are positive real numbers which will be determined later. The function \({\mathscr {L}}_p[u]\) contains all (mixed) multi-variable polynomials of order p with carefully chosen coefficients. Thanks to the non-negativity of the solution, \((\mathscr {L}_p[u])^{1/p}\) is an equivalent \(L^p\)-estimate of u. Certainly, the difficulty is to choose these coefficients so that they are compatible with both diffusion and reactions. In our case, the assumptions on the quasilinear diffusion (Q2) and the intermediate sum condition (A4) allow us construct such an energy function.
Comparing to [12, 22], this work extends this \(L^p\)-energy method to the case of degenerate quasi-linear diffusion coefficients. Moreover, it is also shown that this method is sufficiently robust to model variants and different boundary conditions.
Conditionally, if we can, by using a specific structure, get a better a prior estimate of solutions, the exponent r in the intermediate sum condition (A4) can be enlarged. This is contained in the following theorem.
Theorem 1.3
Assume (1.3), (1.4), (Q1), (Q2), (Q3), (A1), (A2), (A4) and (A5). Suppose that there exists a constant \(a\ge 1\) and \(\mathscr {F}(T)\in (0, \infty )\) such that
and
Then for any nonnegative, bounded initial datum \(u_0\in (L^{\infty }(\Omega ))^m\), there exists a global weak solution to (1.2) with \(u_i\in L^{\infty }_{loc}(0,\infty ; L^{\infty }(\Omega ))\) for all \(i=1,\ldots ,m.\) In particular, if \(\sup _{T\ge 0}\mathscr {F}(T)<\infty \), then the solution is bounded uniformly in time, i.e.
Remark 1.4
When \(b=0\), our results cover the result in [12, Theorem 1.3].
Theorem 1.5
Assume (1.3), (1.4), (Q1), (Q2), (Q3), (A1), (A2), (A4) and (A5). Suppose that there exists a constant \(q> 1\) such that
and
Then for any nonnegative, bounded initial datum \(u_0\in (L^{\infty }(\Omega ))^m\), there exists a global weak solution to (1.2) with \(u_i\in L^{\infty }_{loc}(0,\infty ; L^{\infty }(\Omega ))\) for all \(i=1,\ldots ,m.\) In particular, if \(\sup _{T\ge 0}\mathscr {F}(T)<\infty \), then the solution is bounded uniformly in time, i.e.
Remark 1.6
When \(b=0\), our results cover the result in [12, Theorem 1.3].
It is worth noting that other conditions can also lead to a prior estimates. For example, the entropy condition has been considered in many papers, especially when it involves chemical reactions, see e.g. [3, 5, 29, 30]. This condition means that there exist the scalars \(\mu _i\in \mathbb {R}\) such that
This condition guarantees an \(L^{1}(\Omega )\)-estimate for \(u_i\log u_i\). Actually, this condition guarantees an \(L^{1}(\Omega )\)-estimate for \(H(u(\cdot ,t))\), where
Moreover, we can assume there exists a set \(M=\prod _{k=1}^m(\alpha _i,\beta _i)\), where \(\alpha _i,\beta _i\) are extended real numbers such that \(\alpha _i<\beta _i\) for each \(i=1,\ldots , m\), and solutions to (1.2) remain in M if initial data lies in M, a function \(H:M\rightarrow \mathbb {R}_+\) that is \(C^2\) and has the form
where \(h_i:(\alpha _i,\beta _i)\rightarrow \mathbb {R}_+\) satisfies
for some \(K_5, K_6>0\). Then, we would obtain an \(L^1(\Omega )\)-estimate for \(H(u(\cdot ,t))\) (this has been introduced in [19]). In addition, intermediate sum conditions could also be written in the form
that would lead to results in the same manner as we obtain from (A3) and (A4) above.
Theorem 1.7
Assume (1.3), (1.4), (Q1), (Q2), (Q3), (A1), (A2), (A5), (H1) and (H2). Assume moreover that
Then for any nonnegative, bounded initial datum \(u_0\in (L^{\infty }(\Omega ))^m\), there exists a global weak solution to (1.2) with \(u_i\in L^{\infty }_{loc}(0,\infty ; L^{\infty }(\Omega ))\) for all \(i=1,\ldots ,m.\) In particular, if \(K_5<0\) or \(K_5=K_6=0\), then the solution is bounded uniformly in time, i.e.
Remark 1.8
When \(b=0\), our results cover the result in [12, Theorem 1.2].
Our method is sufficiently robust to extend to other boundary conditions (for example, Robin-type boundary conditions). The precise results are given in the following theorem.
Theorem 1.9
Consider the system
where \(\eta \) is the unit outward normal vector on \(\partial \Omega \), and \(\alpha _i\ge 0\) for all \(i=1,\ldots , m\).
Assume (1.3), (1.4), (Q1), (Q2), (Q3), (A1), (A2), (A4), (A5). Moreover, assume either (A3) or (1.7) or (1.10) with
Then for any non-negative, bounded initial data \(u_0\in L^{\infty }(\Omega )^m\), there exists a global weak solution to (1.15) (see Definition 4.1) with \(u_i\in L^{\infty }_{loc}(0,\infty ;L^{\infty }(\Omega ))\) for all \(i=1,\ldots , m\). In particular, if \(K_1< 0\) or \(K_1= K_2 = 0\) in case of (A3), or \(\sup _{T\ge 0}\mathscr {F}(T)<\infty \) in case of (1.7) or (1.10), then the solution is bounded uniformly in time, i.e.
Remark 1.10
We believe Theorem 1.9 can also be extended to the semilinear boundary conditions of the form
where the nonlinearities \(G_i\) also satisfy a quasi-positivity condition and an intermediate sum condition. The details are left for the interested reader. We refer to [23, 28] for a related work dealing with constant diffusion coefficients.
The paper is organised as follows. In Sect. 2, we prove the existence of Theorem 1.1 by first considering an approximate system where we regularize the nonlinearities to obtain global approximate weak solutions. Then we derive uniform estimates, applying the key idea of the \(L^p\)-energy functions, and pass to the limit to obtain global existence of (1.2). The uniform-in-time boundedness is proved in Sect. 3. The proofs of extended Theorems 1.3, 1.5, 1.7, and 1.9 are given in Sect. 4. In Sect. 5, we show application of our results to a Susceptible-Exposed-Infected-Recovered (SEIRD) model and its variants. Finally, in Appendix A, we give the proof of Lemma 2.9.
Notation. In this paper we use the following notation, some of which will be recalled from time to time:
-
For \(T>0\) and \(p\in [1,\infty ]\), \(Q_T:= \Omega \times (0,T)\) and
$$\begin{aligned} L^p(Q_T):= L^p(0,T;L^p(\Omega )) \end{aligned}$$equipped with the usual norm
$$\begin{aligned} \Vert f\Vert _{L^{p}(Q_T)}:= \left( \int _0^T\int _{\Omega }|f|^pdxdt\right) ^{1/p} \end{aligned}$$for \(1\le p < \infty \) and
$$\begin{aligned} \Vert f\Vert _{L^{\infty }(Q_T)}:= \underset{(x,t)\in Q_T}{\text { ess sup}}|f(x,t)|. \end{aligned}$$ -
For \(p\in [1,\infty ]\), \(\tau \ge 0\) and \(\delta >0\), we denote by
$$\begin{aligned} Q_{\tau ,\tau +\delta }:= \Omega \times (\tau ,\tau +\delta ) \end{aligned}$$and
$$\begin{aligned} L^p(Q_{\tau ,\tau +\delta }):= L^p(\tau ,\tau +\delta ; L^p(\Omega )). \end{aligned}$$
2 Global existence of bounded weak solutions
Definition 2.1
(Weak solutions) A vector of nonnegative state variables \(u=(u_1,\ldots ,u_m)\) is called a weak solution to (1.2) on (0, T) if
where b is in (Q2), and
with \(u_i(\cdot , 0)=u_{i,0}(\cdot )\) for all \(i=1,\ldots ,m\), and for any test function \( {\psi } \in L^2(0,T; H^1(\Omega ))\) we have
2.1 Approximate system
For \(i=1,\ldots ,m\) and \(\varepsilon >0\), consider the following approximating system for \(u^{\varepsilon }=(u_1^{\varepsilon }, \ldots , u_m^{\varepsilon })\),
where
and \(u_{i,0}^{\varepsilon }\in L^{\infty }(\Omega )\) is non-negative and \(u_{i,0}^{\varepsilon }{\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}}u_{i,0}\) in \(L^{\infty }(\Omega )\). With this approximation, it is easy to check that the approximated non-linearities \(f_i^{\varepsilon }\) still satisfy the assumptions (A1)–(A5).
Lemma 2.2
For any fixed \(\varepsilon >0\), there exists a global bounded, nonnegative weak solution to (2.1) on any finite time interval (0, T), \(T>0\).
Proof
Since for fixed \(\varepsilon >0\), the nonlinearities \(f_i^{\varepsilon }(u^{\varepsilon })\) are Lipschitz continuous and bounded, i.e.,
The global existence of a weak solution of (2.1) is standard (see e.g. [18, Chapter VII]).
Next we prove the nonnegativity of \(u^{\varepsilon }\). Denote \(u_{i,+}^{\varepsilon }:=\max \{u_i^{\varepsilon },0\}\) and \(u_{i,-}^{\varepsilon }:=\min \{u_i^{\varepsilon },0\}\). We consider the auxiliary system of (2.1)
where \(u_+^{\varepsilon }=(u_{i,+}^{\varepsilon })_{i=1,\ldots ,m}\).
By multiplying this auxiliary system by \(u_{i,-}^{\varepsilon }\) and using the quasi-positivity assumption (A2) (recall that this property also holds for \(f_i^{\varepsilon }\)), we obtain
where we use \(u_{i,0,-}^{\varepsilon }=0\). Thus, we get \(u_{i,-}^{\varepsilon }=0\) a.e. in \(Q_T\), this shows the desired nonnegativity. \(\square \)
2.2 Uniform-in-\(\varepsilon \) estimate
In this subsection, we prove crucial uniform-in-\(\varepsilon \) estimates for the solution to (2.1). Moreover, we want to emphasize that all the constants in this subsection are independent of \(\varepsilon \).
The following bound in \(L^{\infty }(0,T; L^1(\Omega ))\) is immediate.
Lemma 2.3
Assume (A1), (A2) and (A3). Then for any \(T>0\), there exists a constant \(M_T\) depending on \(T, \Omega , \Vert u_{i,0}\Vert _{L^1(\Omega )}\) and \(c_1,\ldots ,c_m\), \(K_1, K_2\) in (A3) such that
Proof
By summing the equation in (2.1), integrating on \(\Omega \) and using (A3) we have
The classical Gronwall inequality gives the desired estimate. \(\square \)
The following \(L^p\)-energy function has been developed in [12, 23]. We write \(\mathbb {Z}_{+}^m\) as the set of all m tuples of nonnegative integers. Addition and scalar multiplication by nonnegative integers of elements in \(\mathbb {Z}_{+}^m\) is understood in the usual manner. If \(\beta =\left( \beta _1, \ldots , \beta _m\right) \in \mathbb {Z}_{+}^m\) and \(p \in \mathbb {Z}_+\), then we define \(\beta ^p=\left( \left( \beta _1\right) ^p, \ldots ,\left( \beta _m\right) ^p\right) \). Also, if \(\alpha =\left( \alpha _1, \ldots , \alpha _m\right) \in \) \(\mathbb {Z}_{+}^m\), then we define \(|\alpha |=\sum _{i=1}^m \alpha _i\). Finally, if \(z=\left( z_1, \ldots , z_m\right) \in \mathbb {R}_{+}^m\) and \(\alpha =\) \(\left( \alpha _1, \ldots , \alpha _m\right) \in \mathbb {Z}_{+}^m\), then we define \(z^\alpha =z_1^{\alpha _1} \cdots z_m^{\alpha _m}\), where we interpret \(0^0\) to be 1. For any \(2\le p \in {\mathbb {N}}\), we build our \(L^p\)-energy function of the form
where
with
and \(\theta =\left( \theta _1, \ldots , \theta _m\right) \), where \(\theta _1, \ldots , \theta _m\) are positive real numbers which will be determined later. For \(p=0,1,2\), one can write these functions explicitly as
and
Thanks to the nonnegativity of the solution, we have
This will allow us to use \(\mathscr {L}_p[u](t)\) to obtain a priori estimates on u for each \(2 \le p \in \mathbb {N}\). We will need two technical lemmas.
Lemma 2.4
([12], Lemma 4.1) Suppose \(m \in \mathbb {N}, \theta =\left( \theta _1, \ldots , \theta _m\right) \), where \(\theta _1, \ldots , \theta _m\) are positive real numbers, \(\beta \in \mathbb {Z}_{+}^m\), and \(\mathscr {H}_p[u]\) is defined in (2.3). Then
and for \(p \in \mathbb {N}\) such that \(p \ge 2\),
Lemma 2.5
([12], Lemma 4.2) Suppose \(m \in \mathbb {N}, \theta =\left( \theta _1, \ldots , \theta _m\right) \), where \(\theta _1, \ldots , \theta _m\) are positive real numbers, and let \(\mathscr {H}_p[u]\) be defined in (2.3). If \(p \in \mathbb {N}\) such that \(p \ge 2\), then
where
Next, we need the following functional inequality.
Lemma 2.6
Suppose \(\Omega \subset \mathbb {R}^N\) such that the Gagliardo-Nirenberg inequality is satisfied and basic trace theorems apply (for instance \(\Omega \) has a Lipschitz boundary). Let \(a\ge 1\), \(p\ge 2a\), \(w: \overline{\Omega }\rightarrow \mathbb {R}_+\) such that \(w^{\frac{b+p}{2}}\in W^{1,2}(\Omega )\) and there exists \(K\ge 0\) such that \(\Vert w\Vert _{L^a(\Omega )}\le K\). If \(1\le r<1+b+ \frac{2a}{N}\) and \(b\ge 0\), then there exists \(C_{\varepsilon }\ge 0\) (depending on \(p,\varepsilon , r, a, b, K, \Omega ,\) but independent of w) such that
Proof
By using Sobolev’s embedding we have
Thus we have
Since
if \(r<b+1\). Then we have
Next we consider \(r\ge b+1\). Thanks to the Gagliardo-Nirenberg inequality, we have
where
It follows that
and
Since \(1\le r<1+b+\frac{2a}{N}\), we find
We can use Young’s inequality to estimate
where
If \(b+p=2a\), then this term is bounded by a constant depending on K, since \(\Vert w\Vert _{L^a(\Omega )}\le K\). If \(b+p>2a\), we use an interpolation inequality to have
where \(\theta \in (0,1)\) satisfies
Note that
due to \(r<b+1+ \frac{2a}{N}\) and \(b\le r-1<2r\).
From (2.8)–(2.10) and Young’s inequality, it follows that
Inserting this into (2.6), we get
Combine this with (2.4) and (2.5) leads to the desired estimate
\(\square \)
The following lemma shows results for the intermediate sum condition (A4), which is crucial for constructing the \(L^p\) energy function.
Lemma 2.7
([12], Lemma 2.4) Assume (A4). Then there exist componentwise increasing functions \(g_i: \mathbb {R}^{m-i} \rightarrow \mathbb {R}_{+}\) for \(i=1, \ldots , m-1\) such that if \(\theta =\left( \theta _1, \ldots , \theta _m\right) \in (0, \infty )^m\) satisfies \(\theta _m>0\) and \(\theta _i \ge g_i\left( \theta _{i+1}, \ldots , \theta _m\right) \) for all \(i=1, \ldots , m-1\), then
for some constant \(K_\theta \) depending on \(\theta , g_i\), and \(K_3\) in (A4).
We now use the \(L^p\)-energy functions to obtain the \(L^p\)-estimates of \(u^{\varepsilon }\).
Lemma 2.8
Assume (1.3), (1.4), (Q1), (Q2), (A1), (A2), (A3) and (A4) with \(1\le r<1+b+\frac{2}{N}\). Then for any \(1 \le p<\infty \) and any \(T>0\), there exists a constant \(C_{T, p}\) depending on T, p and other parameters such that
Proof
Let \(u^{\varepsilon }\) solve (2.1), and \(\mathscr {L}_p(t):=\mathscr {L}_p\left[ u^{\varepsilon }\right] (t)\) be defined in (2.2). Then
For (I), we apply Lemma 2.5 and integration by parts, we have
with
For a given \(\beta \) with \(|\beta |=p-2\), create an \(m N \times m N\) matrix \(B(\beta )\) made up of \(m^2\) blocks \(B_{k, l}(\beta )\), each of size \(N \times N\), where
Note that for each \(k=1, \ldots , m\),
Also,
where \(\nabla u^{\varepsilon }(x, t)\) is a column vector of size \(m N \times 1\), and for \(j=1, \ldots , m\), entries \(N(j-1)+1\) to Nj of \(\nabla u^{\varepsilon }(x, t)\) are \(\nabla u_j^{\varepsilon }(x, t)\). We claim that if all of the entries in \(\theta \) are sufficiently large, then \(B(\beta )\) is positive definite. In fact, it is a simple matter to show it is positive definite if and only if the \(m N \times m N\) matrix \({\widetilde{B}}(\beta )\) made up of \(N \times N\) blocks
is positive definite. However, if we recall the uniform positive definiteness of the matrices \(D_k\), we can show that if \(\theta _i\) is sufficiently large for each i, then we have what we need. Consequently, returning to the above, we can show there exists \(\alpha _p>0\) so that
where we used (Q2).
From (A4) and Lemma 2.7, we choose the components of \(\theta = (\theta _i)\) inductively so that \(\theta _i\) are sufficiently large that the previous positive definiteness condition of \({\widetilde{B}}_{k, l}(\beta )\) is satisfied, and
where \(g_i\) are functions constructed in Lemma 2.7. Note that \(\theta _i \le \theta _i^{2\beta _i + 1} \le \theta _i^{2p - 1}\). Since \(g_i\) is componentwise increasing, the relation (2.13) implies
Now we can apply Lemma 2.7, to obtain some \(K_{\widetilde{\theta }}\) so that for all \(\beta \in \mathbb {Z}_+\) with \(|\beta |=p-1\), we have
It follows that there exists \(C_p>0\), such that (2.12) implies
Thus, we have
Combining Lemma 2.6, we get
This implies
Thus, we have
This finishes the proof of Lemma 2.8. \(\square \)
Lemma 2.9
Assume (1.3), (1.4), (Q1), (Q2), (A1), (A2), (A3), (A5) and (A4) with \(1\le r<1+b+\frac{2}{N}\). Then for any \(T>0\), the solution of (2.1) is bounded in \(L^{\infty }\) in time, i.e.,
for some constant \(C_T\) depending on T and independent of \(\varepsilon >0\).
Proof
The proof is similar to the proof of Lemma 2.3 in Ref. [9]. For the convenience of reading, we give the specific proof process in the Appendix A. \(\square \)
2.3 Passing to the limit—Global existence
Lemma 2.10
Assume (1.3). For any \(k>0\), we have
Proof
Let \(T_k(z)\) be defined
Define \(S_k(z)=\int ^z_0 T_k(\tau ) \,d\tau \) and multiply (2.1) by \(T_k(u^{\varepsilon }_i)\) we obtain
By applying the properties of \(T_k\) and \(S_k\), the right hand is bounded by
From (1.3) and \( \nabla (T_k(u^{\varepsilon }_i))=\chi _{\{|u^{\varepsilon }_i|\le k\}} \nabla u^{\varepsilon }_i\), it follows that
Finally, by using \(S_k(u^{\varepsilon }_i)\ge 0\) we obtain our desired estimate. \(\square \)
Lemma 2.11
Assume (1.3). For any \(\beta >0\), there exists constants C such that
Proof
Let \(M:=\Vert u_0\Vert _{L^{1}(\Omega )}+\Vert f^{\varepsilon }_i(u^{\varepsilon })\Vert _{L^{1}(Q_T)}\), we can apply Lemma 2.10 to obtain
Thus (2.16) holds for \(C:=\frac{2}{\lambda } \sum ^{\infty }_{j=0}(2^{-\beta })^{j}\), which is finite since \(\beta >0\). \(\square \)
Proof of Theorem 1.1-Global existence
From Sect. 2.2 we have the following bound
Due to the polynomial growth (A5), for any \(i=1, \ldots , m\), we can get
By multiply (2.1) by \(u_i^{\varepsilon }\) then integrating on \(Q_T\) gives
By (1.3), (Q2) and Hölder inequality give
In particular
By testing the equation (2.1) with \(\varphi \in L^2(0,T; H^1(\Omega ))\), we have
where we use (1.4), (Q3), (2.16) and (2.17). Thus we can get
From Lemma 2.9, (2.18) and (2.19), we can apply a nonlinear version of the well known Aubin-Lions Lemma (see e.g. [21, Theorem 1.1]) to ensure
Thanks to the \(L^{\infty }\) bound of Lemma 2.9, this convergence in fact holds in \(L^{p}(Q_T)\) for any \(1\le p<\infty \).
Since the sequence \((u_i^\varepsilon )^{\frac{b+2}{2}}\) is uniformly bounded in \(L^2(0,T; H^{1}(\Omega ))\), there exists a subsequence (still denoted by \((u_i^\varepsilon )\)) and \(v\in L^2(0,T; H^{1}(\Omega ))\) such that
Now, we are ready to identify the weak limit v by the following computation: for testing functions \(\varphi \in C^{\infty }_0(Q_T)\) we have
This implies that the function \(u_i^{\frac{b+2}{2}}\) is weakly differentiable with weak derivative v, i.e., \( u_i^{\frac{b+2}{2}} = v\). Thus
It remains to pass to the limit \(\varepsilon \rightarrow 0\) in the weak formulation of (2.1) with \(\eta \in L^2(0,T; H^1(\Omega ))\),
The convergence of the first term on the left hand side and the term on the right hand side of (2.22) is immediate.
The convergence
follows from \(\frac{\Phi (u^{\varepsilon })}{(u^{\varepsilon }_i)^{\frac{b}{2}}} {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \frac{\Phi (u)}{u_i^{\frac{b}{2}} } \) a.e. in \(Q_T\), Lemma 2.9 and the equation (2.21). Passing to the limit in the weak formulation (2.22), we have
We obtain that \(u=\left( u_1,\ldots ,u_m\right) \) is a global weak solution to (1.2) and additionally
\(\square \)
3 Uniform-in-time boundness
Lemma 3.1
Assume (A1), (A2) and (A3) with either \(K_1<0\) or \(K_1=K_2=0\). Then, there exists a constant M independent of time such that
Proof
Similar to the proof of Lemma 2.3, we have
Integrating over (s, t) we have
for all \(t>s\ge 0\).
If \(K_1=K_2=0\), the bound (3.1) follows immediately.
If \(K_1<0\), we get for some constant \(\sigma >0\) and for all \(t>s\ge 0\),
Define
It follows from (3.2) that
which leads to
Integrating with respect to s on (0, t), and using \(\phi (t)=0\), we have
Since \(\sigma \phi (0)-K_2|\Omega | t \le \sum _{i=1}^m c_i \int _{\Omega }u_{i 0}(x)\,d x\) (see (3.2)) it follows that
which finishes the proof of Lemma 3.1. \(\square \)
Proof of Theorem 1.1-uniform-in-time boundness
We first show that for any \(1\le p<\infty \), there exists a constant \(C_p>0\) such that
Indeed, by using the \(L^p\)-energy function \(\mathscr {L}_p[u]\) defined in (2.2) and similar to Lemma 2.8, we obtain
Now, we apply Lemma 2.6 to the right hand side, and it is note that the \(L^1\)-bound is uniform in time (due to Lemma 3.1), we get
This implies
Thus, we have
Next we will show that the solution is bounded uniformly in time in sup norm. We use a smooth time-truncated function \(\psi : \mathbb {R}\rightarrow [0,1]\) with
and \(0\le \phi '\le C\), and its shifted version \(\psi _{\tau }(\cdot )=\psi (\cdot ,-\tau )\) for any \(\tau \in \mathbb {N}\). Let \(\tau \in \mathbb {N}\) be arbitrary. It is straightforward to show that since \(u=(u_i)_{i=1,\ldots ,m}\) is a weak solution to (1.2), the function \(\psi _{\tau }u=(\psi _{\tau }u_i)_{i=1,\ldots ,m}\) is a weak solution to the following
Thanks to (3.3) and the polynomial growth (A5), we have
where \(1\le p<\infty \). Therefore, similar to the proof of Lemma 2.9, we get
where C is a constant independent of \(\tau \in \mathbb {N}\). Thanks to \(\psi _{\tau }\ge 0\) and \(\psi |_{(\tau +1,\tau +2)}\equiv 1\), we obtain finally the uniform-in-time bound
and the proof of Theorem 1.1 is complete. \(\square \)
4 Proof of Theorems 1.3–1.9
Proof of Theorem 1.3
For the global existence, we only need to show that for any \(1\le p<\infty \) and any \(T>0\), there exists \(C_{T,p}>0\) such that
The rest follows exactly as in Lemma 2.9. To show (4.1), we use the \(L^p\)-energy functions \(\mathscr {L}_p(t)\) constructed in Lemma 2.8 until (2.14) we end up with
Since (1.7) and (1.8), we apply Lemma 2.6 to estimate
where \(C_{T}\) depends on \(\mathscr {F}(T)\). Inserting this into (4.2) yields
which implies (4.1).
For the uniform-in-time bounds, we use an argument similar to the proof of Theorem 1.1. \(\square \)
Proof of Theorem 1.5
For the global existence, we only need to show that for any \(1\le p<\infty \) and any \(T>0\), there exists \(C_{T,p}>0\) such that
The rest follows exactly as in Lemma 2.9. To show (4.3), we use the \(L^p\)-energy functions \(\mathscr {L}_p(t)\) constructed in Lemma 2.8 until (2.14) we obtain
We integrate (4.4) in time to obtain
Denote by \(y_i:=u_i^{\varepsilon }(x,t)^{\frac{b+p}{2}}\). The left-hand side (LHS) of (4.5) can be estimated below by
For the right-hand side of (4.5), we first consider
as a constant to be determined later. Of course we are only interested in the case when \(p-1+r>q\), otherwise the right hand side of (4.5) is bounded thanks to (1.10). By the interpolation inequality we have
where \(\theta \in (0,1)\) satisfies
which implies
Using this, and taking into account (1.10), (4.8) implies
For p large enough, we choose \(\vartheta \) close enough to (but bigger than) \(p-1+r\) so that \(H^1(\Omega )\hookrightarrow L^{\frac{2\vartheta }{b+p}}(\Omega )\). That means \(\vartheta \) is arbitrary for \(N\le 2\) and
Thus, we can use the Gagliardo-Nirenberg’s inequality to estimate
where \(\alpha \in (0,1)\) satisfies
From this
Therefore, we obtain from (4.11) that
It follows that
We choose
which is possible since \(p - 1 + r < p + b+ \frac{2p}{N}\) for p large enough. Thus, by Hölder’s inequality,
Inserting this into (4.9) yields
where \(\theta \in (0,1).\)
Using Young’s inequality of the form
we can choose \(\vartheta \) such that
This is equivalent to
We now check that we can choose \(\vartheta \) which satisfies all the conditions (4.7), (4.10), (4.12) and (4.15). This is fulfilled provided
and this can be obtain from our assumption (1.11). We estimate (4.13) further as
Thus
Combining this with (4.6) gives us the desired estimate (4.3).
For the uniform-in-time bounds, We use an argument similar to the proof of Theorem 1.1. \(\square \)
Proof of Theorem 1.7
We give a formal proof as its rigor can be easily obtained through approximation. Since \(h_k(\cdot )\) is convex, we have
Therefore, by defining \(v_i:= h_i(u_i)\ge 0\) we have
with initial data \(v_i(x,0) = h_i(u_{i,0}(x))\) and homogeneous Dirichlet boundary condition \(v_i(x,t) = 0\) for \(x\in \partial \Omega \) and \(t>0\). Thanks to (H1) and we have
and
We can now reapply the methods in the proof of Theorem 1.1 to obtain \(v_i \in L^\infty _{\text {loc}}(0,\infty ;L^{\infty }(\Omega ))\), and in case \(K_5<0\) or \(K_5 = K_6 = 0\) in (H1),
Due to the assumption (H1), the global existence and boundedness of (1.2) immediately hold. \(\square \)
Definition 4.1
A vector of non-negative concentrations \(u = (u_1, \ldots , u_m)\) is called a weak solution to (1.15) on (0, T) if
with \(u_i(\cdot ,0)= u_{i,0}(\cdot )\) for all \(i=1,\ldots , m\), and for any test function \(\varphi \in L^2(0,T;H^1(\Omega ))\) with \(\partial _t\varphi \in L^2(0,T;H^{-1}(\Omega ))\), it holds that
Proof of Theorem 1.9
The proof of this theorem is similar to that of Theorems 1.1, 1.3 and 1.5, except for the fact that the \(L^1\)-norm can be obtained in a different, and easier, way. Since \(\varphi \equiv 1\) is an admissible test function we get from (4.17) and (A3) that
The \(L^1\)-bound is uniform in time in case \(K_1<0\) or \(K_1 = K_2 =0\) (using a similar idea to Lemma 3.1). \(\square \)
5 Applications
We show the application of our results to SEIR(-D) model considered in [1, 31, 32] and its variants. First, the SEIR model introduced in [1, 31, 32] reads as
subject to homogeneous Neumann boundary conditions \(\nabla s \cdot \eta =\nabla e \cdot \eta =\nabla i \cdot \eta =\nabla r \cdot \eta =0\) on \(\partial \Omega \), where the symbols s, e, i and r denote the susceptible-, exposed-, infected-, recovered populations, respectively, and \(n:=s+e+i+r\) is the total living population, \(\nu _s\), \(\nu _e\), \(\nu _i\), \(\nu _r\), \(\beta _i\), \(\beta _e\), \(\alpha \), \(\mu \), \(\phi \), \(\sigma \), \(\phi _d\), \(\phi _r\) and \(\phi _e\) are positive constants. The global existence of bounded weak solutions was shown in [1] where it was imposed that all diffusion rates are the same, i.e. \(\nu _s = \nu _e = \nu _i = \nu _r\), the term \(1-A_0/n\) is replaced by a non-singular function A(n), for instance \(A(n) = (1-A_0/n)_+\), and the term \(-\phi _dni\) is replaced by \(-\phi _d i\). These are needed to show the total population n is pointwise bounded from below by a positive constant, making the diffusion operators non-degenerate.Footnote 3 The method therein seems to break down when the diffusion are different. By applying our main results in Theorem 1.1, we show that these assumptions can be removed. We have the following result.
Theorem 5.1
Assume all diffusion coefficients \(\nu _s, \nu _e, \nu _i, \nu _r\), and other parameters in (5.1) to be positive. Then, for any non-negative and bounded initial data \((s_0,e_0,i_0,r_0)\in L_+^{\infty }(\Omega )^4\), there is a global bounded weak solution to (1.1), i.e. for any \(T>0\),
Moreover, if \(\alpha \le \mu \), then this solution is bounded uniformly in time, i.e.
Proof
It is noted that the nonlinearities in (5.1) are not locally Lipschitz continuous around 0. We circumvent this problem by considering the following approximating system for each \(\delta >0\),
subject to homogeneous Neumann boundary conditions \(\nabla s^{\delta } \cdot \eta =\nabla e^{\delta } \cdot \eta =\nabla i^{\delta } \cdot \eta =\nabla r^{\delta } \cdot \eta =0\) on \(\partial \Omega \), where \(u^\delta = (s^\delta , e^\delta , i^\delta , r^\delta )\), \(n^\delta = s^\delta + e^\delta + i^\delta + r^\delta \), \(f^{\delta }_s, f^{\delta }_e, f^{\delta }_i\) and \(f^{\delta }_r\) the nonlinearities in the equations of \(s^{\delta }, e^{\delta }, i^{\delta }\) and \(r^{\delta }\), respectively. Consequently
Similar to the proof of Theorem 1.1, we can get
where \(C_T\) independent of \(\delta \). This independence on \(\delta \) can be obtained since the constant \(C_T\) obtained in Theorem 1.1 depends only on the size of initial data, domain \(\Omega \), diffusion coefficients, and the parameters in the assumptions (A1)–(A5). Due to the polynomial growth of \(f^{\delta }_s, f^{\delta }_e, f^{\delta }_i\) and \(f^{\delta }_r\), we can get
By multiply the first equation of (5.4) by \(s^{\delta } \) then integrating on \(\Omega _T\) gives
Since \(n^\delta \ge s^\delta \), it follows
By testing the first equation of (5.4) with \(\phi \in L^2(0,T; H^1(\Omega ))\), we have
where we used (5.5) and (5.6). Thus,
Applying a nonlinear version of Aubin-Lions Lemma, see e.g. [21, Theorem 1] to ensure
Thanks to the \(L^{\infty }\) bound of \(s^{\delta }\), this convergence in fact holds in \(L^{p}(Q_T)\) for any \(1\le p<\infty \). From the inequality (5.7), the sequence \((s^{\delta })^{\frac{3}{2}}\) is uniformly bounded in \(L^2(0,T; H^{1}(\Omega ))\), there exists a subsequence (still denoted by \((s^{\delta })\)) and \(v\in L^2(0,T; H^{1}(\Omega ))\) such that
Now, we are ready to identify the weak limit v by the following computation: for testing functions \(\varphi \in C^{\infty }_0(Q_T)\) we have
This implies that the function \(s^{\frac{3}{2}}\) is weakly differentiable with weak derivative v, i.e., \( s^{\frac{3}{2}} = v\). Thus
Similarly, we can get the strong convergence of \((e^{\delta }, i^{\delta }, r^{\delta })\) in \(L^{p}(Q_T)^3\) for any \(1\le p<\infty \), and the weak convergence of \(((e^{\delta })^{\frac{3}{2}}, (i^{\delta })^{\frac{3}{2}}, (r^{\delta })^{\frac{3}{2}})\) in \(L^2(0, T; H^1(\Omega ))^3\). Thus, \(n^{\delta } = s^\delta + e^\delta + i^\delta + r^\delta \) converges strongly to n in \(L^{p}(Q_T)\) for any \(1\le p<\infty \). Now, let \(\delta \rightarrow 0\) in the weak formulation of (5.4) with \(\psi \in L^2(0, T; H^1(\Omega ))\) and using [7, Lemma A.2], we get the weak solution \(u= (s, e, i, r)\) of (5.1).
Furthermore, if \(\alpha \le \mu \), we have the uniform-in-time boundedness first for solutions to (5.4) and consequently for (5.1). \(\square \)
Thanks to the robustness of our approach, we can also consider a variant (5.1) where
-
\(\beta _i\) and \(\beta _e\) are functions of n or the combinations of n and species i or e (see the original model proposed in [32]);
-
the diffusion rates \(\nu _s, \nu _e, \nu _i, \nu _r\) are functions of (x, t), which shows the possibly high heterogeneity of the environment; and
-
and the other rates \(\alpha , \mu , \sigma , \phi _e, \phi _d, \phi _r\) are non-negative functions of (x, t),
and therefore the system reads as
subject to homogeneous Neumann boundary conditions \( \nabla s \cdot \eta =\nabla e \cdot \eta =\nabla i \cdot \eta =\nabla r \cdot \eta =0\). We assume the following:
-
(M1’)
\(\beta _i(i,n), \beta _e(e,n)\) are non-negative continuous functions and satisfy
$$\begin{aligned} \beta _i(i,n) \le c_i \left( 1 + i + n\right) , \quad \beta _e(e,n) \le c_e\left( 1 + e+n\right) , \end{aligned}$$for some constants \(c_i, c_e>0\),
-
(M2’)
for \(z\in \{s,e,i,r\}\), \(\nu _z: \Omega \times [0,\infty ) \rightarrow \mathbb {R}^{n\times n}\) such that \(\nu _z\in L^\infty (\Omega \times (0,T);\mathbb {R}^{n\times n})\) for each \(T>0\), and there is a constant \(\lambda _z>0\) such that
$$\begin{aligned} \lambda _z|\xi |^2 \le \xi ^\top \nu _z(x,t) \xi , \quad \forall (x,t)\in \Omega \times [0,\infty ), \quad \forall \xi \mathbb {R}^n; \end{aligned}$$ -
(M3’)
all non-negative functions \(\alpha , \mu , \sigma , \phi _e, \phi _d, \phi _r: \Omega \times [0,\infty )\) are bounded by a common constant, i.e. \(\exists M>0\) such that
$$\begin{aligned} (\alpha + \mu + \sigma + \phi _e + \phi _d+\phi _r)(x,t) \le M, \quad \forall (x,t)\in \Omega \times [0,\infty ). \end{aligned}$$
The following theorem follows directly from Theorem 1.1.
Theorem 5.2
Assume (M1’)–(M3’). Then, for any non-negative bounded initial data, there exists a global bounded weak solution to (5.10), i.e. for any \(T>0\),
In particular, if \(\alpha (x,t)\le \mu (x,t)\) for all \((x,t)\in \Omega \times [0,\infty )\), then the solution is bounded uniformly in time, i.e.
Data Availability Statement
No datasets were generated or analysed during the current study.
Notes
In fact, [1] generalised the diffusion rate to \(\kappa (n)\) for a continuous function \(\kappa (\cdot )\), but it does not alter the fact that the diffusion rates must be the same so that the technique therein can be applied.
In fact, thanks to the lower bound of n, it was possible to replace all diffusion operators by a common function \(\kappa (n)\), where \(\kappa : (0,\infty )\rightarrow (0,\infty )\) is a continuous function.
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Acknowledgements
The authors would like to thank Prof. Pierluigi Colli, Prof. Gabriela Marinoschi, and Prof. Elisabetta Rocca for their inspired and fruitful discussion. This research was funded in whole, or in part, by the Austrian Science Fund (FWF) [I 5213]. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
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The authors would like to thank Prof. Pierluigi Colli, Prof. Gabriela Marinoschi, and Prof. Elisabetta Rocca for their inspired and fruitful discussion. J. Yang is supported by NSFC Grants No. 12271227 and China Scholarship Council (Contract No. 202206180025). B.Q. Tang received funding from the FWF project “Quasi-steady-state approximation for PDE”, number I-5213.
Proof of Lemma 2.9
Proof of Lemma 2.9
We first state the following interpolation inequality.
Lemma A.1
Let \(\gamma \ge 1\) and \(\alpha >0\). Then we have the following interpolation inequality
where
Proof
For \(0<\beta <\infty \). We apply an interpolation inequality to get
with \(\theta \in (0,1)\) satisfying
It follows that
From (A.3) and \(\lambda >0\) to be chosen, using Young’s inequality we obtain
Now, we choose \(\lambda \) such that
which is equivalent to
where we used (A.5). Therefore
with
For \(\beta =\infty \), we can use a similar method to get the result. This completes the proof of Lemma A.1. \(\square \)
Lemma A.2
([10], Lemma 2.4) Let \(\left\{ y_{n}\right\} _{n \ge 1}\) be a sequence of positive numbers which satisfies
where \(K, B>0\) and \(\gamma , \kappa >1\) are independent of n. Then there exists \(\varepsilon >0\) such that, if \(y_{1} \le \varepsilon \), then
We are now presenting a proof of Lemma 2.9 using the idea of Moser iteration.
Proof of Lemma 2.9
From (A5),
Using Lemma 2.8, we can get for any \(1\le p<\infty \) a constant \(C_p>0\) exists depending on p (but not on T) such that
Let \(k \ge 1\) be a constant which will be specified later. For each \(j \ge 0\), we define
and
The following simple observations will be helpful
By multiplying the equation \(\partial _t u^{\varepsilon }_i - \Delta \left( D_i(x,t) \,\Phi (u^{\varepsilon })\, \nabla u^{\varepsilon }_i\right) =f^{\varepsilon }_i\) by \(v_{j+1}\) and integrating on \(Q_T\), we have
where we used (1.3). Using (Q2), we have
Note that \(u^{\varepsilon }_i \ge k-\frac{k}{2^j} \ge \frac{k}{2}\) on \(A_j\) (\(j\ge 1\)), we have
thanks to \(k\ge 1\) and the fact that \(v_{j+1} \equiv 0\) on \(Q_T \backslash A_{j+1} \supset Q_T \backslash A_j\) since \(A_{j+1} \subset A_j\). By adding \(\frac{\lambda M_i}{2^{b-1}} \int _0^T\Vert v_{j+1}\Vert ^2_{L^{2}(\Omega )} d t\) to both sides of (A.8), we get
which yields
By definition, when we choose \(k \ge 2\Vert u^{\varepsilon }_0\Vert _{L^{\infty }(\Omega )}\), we have
By using (A.7), we have with \(1 \le \frac{2^{j+1}}{k} v_j\) on \(A_{j+1}\)
Choose \(p>\frac{N+2}{2}\), we have
Moreover,
implying
We now can use Hölder’s inequality to estimate with (A.7)
Inserting (A.10), (A.11) and (A.12) into (A.9) leads to
for all \(j \ge 0\), where \(B=\max \left\{ 2^{\frac{4 }{N}}, 2^{\sigma -1}\right\} \).
By setting \(Y_j=\Vert v_j\Vert _{W(0, T)}^2\), we obtain a sequence \(\left\{ Y_n\right\} _{n \ge 1}\) satisfying the property in Lemma A.2. It remains to show that \(Y_1\) is small enough.
We show now that for any \(\eta >0\), there exists \(k \ge \max \left\{ 1, 2\Vert u^{\varepsilon }_0\Vert _{L^{\infty }(\Omega )}\right\} \) large enough such that
Let \(p> 1\). By multiplying (2.1) by \(p |u_i^{\varepsilon }|^{p-1}\) and integrating over \(\Omega \), we obtain
Integration by parts and the homogeneous Neumann boundary condition \((D(x,t) \Phi (u^{\varepsilon }) \nabla u_i^{\varepsilon })\cdot \nu = 0\) lead to
where we used (1.3) and (Q2). By Hölder’s inequality
Therefore, it follows from (A.15) that
By applying for \(r<1\) the elementary inequality
to (A.16) with \(r=1/p\) and \(y(t) = \Vert u_i^{\varepsilon }(t)\Vert _{L^{p}(\Omega )}^{p}\), we obtain
That means
where \(C_{T}\) defined in (A.17) grows at most polynomially in T. By integrating (A.16) with respect to t on (0, T) and by using Young’s inequality and the convention \(r:= b+p\ge p>1\), we get
By adding \(C(p,\lambda , M_i)\int ^T_0\int _{\Omega }(u_i^{\varepsilon })^{r} \,d x d t\) to both sides, we have
By the Sobolev’s embedding, we have
with
On the other hand, by using the bound \(\Vert u_i^{\varepsilon }(t)\Vert _{L^{p}(\Omega )}^{p} \le C_{T}\) in (A.18) and the interpolation inequality
with
we estimate in the cases \(b>0\) for which \(\alpha <1\)
where we have used Young’s inequality (with the exponents \(1=(1-\alpha ) + \alpha \)) in the last step. Note that if \(b=0\), the bound (A.21) still holds true without the first term and with \(\frac{r}{p}=1\). Inserting (A.20) and (A.21) into (A.19) leads to
It follows that
with
and
with \(D_{T}\) defined in (A.22).
Thus, we have
where \(r=b+p \ge p\) and \(s=\frac{r N}{N-2}\) if \(N \ge 3\) and \(r<s<+\infty \) arbitrary if \(N \le 2\).
Using Lemma A.1, we have
Direct calculations show that \(\tau >2+\frac{4}{N}\) if \(N \ge 2\) and \(\tau >3\) if \(N=1\). In particular,
From (A.9),
Since \(k \ge 2\Vert u^{\varepsilon }_0\Vert _{L^{\infty }(\Omega )},\Vert v_1(0)\Vert ^2_{L^{2}(\Omega )}=\Vert \left( u^{\varepsilon }_0-\frac{k}{2}\right) _+\Vert ^2_{L^{2}(\Omega )}=0\).
Consider now the case \(N \ge 2\). By using (A.7), it yields
recalling that \(v_0=(u^{\varepsilon }_i)_{+}\). Similarly to (A.12), we get
From (A.24), (A.25) and (A.26), we get (A.14) if
Thus, with this choice of k, it follows that
and hence,
which is our desired estimate.
The proof for the case \(N=1\) is very similar using
and
where \(\xi =\frac{1}{2}(2p-3)>0\). We therefore omit the details.
Thus, we completed the proof of Lemma 2.9. \(\square \)
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Yang, J., Morgan, J. & Tang, B.Q. On quasi-linear reaction diffusion systems arising from compartmental SEIR models. Nonlinear Differ. Equ. Appl. 31, 98 (2024). https://doi.org/10.1007/s00030-024-00985-w
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DOI: https://doi.org/10.1007/s00030-024-00985-w