Convergence of a class of nonlinear time delays reaction-diffusion equations

Abstract

Stability under a variational convergence of nonlinear time delays reaction-diffusion equations is discussed. Problems considered cover various models of population dynamics or diseases in heterogeneous environments where delays terms may depend on the space variable. As a consequence a stochastic homogenization theorem is established and applied to vector disease and logistic models. The results illustrate the interplay between the growth rates and the time delays which are mixed in the homogenized model.

Appendices

Appendix A. Appplication of Theorem 5.1 to Homogenization of vector disease models

In the context of vector disease models illustrated in Example 3.1, we consider the delays reaction-diffusion problem modeling the evolution of the density $u_\varepsilon (\omega ,\cdot )$ of some infected population during a time $T>0$, in a $C^1$-regular domain $\Omega$. We assume that $\Omega$ is included in a $\varepsilon$-random checkerboard-like environment, or in an environment made up of spherical heterogeneities of size $\varepsilon$, independently randomly distributed with a given frequency following a Poisson point process (see [2, Appendix B] for a complete description). Recall that the dynamical system $(\Sigma , {\mathcal A}, \mathbf {P}, (T_z)_{z\in \mathbb {Z}^N})$ modeling these two situations is ergodic. We assume that each two functions $a(\omega ,\cdot ,\cdot )$ and $b(\omega , \cdot , \cdot )$ belong to $W^{1,2}(0,T, L^2_{loc}(\mathbb {R}^N))\cap L^\infty ([0,T]\times \mathbb {R}^N)$. When no infected population is located at the boundary, and the history function $\eta _\varepsilon$ is deterministic and satisfies $0\le \eta _\varepsilon \le \overline{\rho }$ for some $\overline{\rho }\ge 1$, the density $u_\varepsilon (\omega ,\cdot )$ is the unique solution of

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle \frac{ du_\varepsilon }{dt}(\omega ,t)-\mathrm{div}D_\xi W\left( \omega , \frac{x}{\varepsilon }, \nabla u_\varepsilon (\omega , t)\right) = a\left( \omega , t,\frac{\cdot }{\varepsilon }\right) ( 1-u_\varepsilon (\omega ,t)) \int _{-\infty }^t u_\varepsilon (\omega ,s) d{\mathbf {m}}_t^\varepsilon (s)\\ \qquad \qquad \qquad \qquad \displaystyle - b\left( \omega , t,\frac{\cdot }{\varepsilon }\right) u_\varepsilon (\omega ,t) \quad \hbox { for a.e. } t\in (0 , T), \\ u_\varepsilon (\omega ,t)= \eta _\varepsilon (t), \quad 0\le \eta _\varepsilon (t)\le \overline{\rho } \hbox { for all } t\in (-\infty , 0],\\ u_\varepsilon (\omega ,t)\in H^1(\Omega ), \ \hbox { div} D_\xi W(\omega , \frac{\cdot }{\varepsilon }, \nabla u_\varepsilon (\omega ,t))\in L^2(\Omega )\hbox { for all } t\in ]0,T], \\ u_\varepsilon (\omega , t) = 0 \hbox { on } \partial \Omega ,\hbox { for all } t\in [0,T], \end{array} \right. \end{aligned}$$

and satisfies $0\le u_\varepsilon (\omega )\le \overline{\rho }$. We are in the context of Theorem 5.1, with $l=2$ and

$$\begin{aligned} r(\omega ,t,x)=(a(\omega ,t,x), - b(\omega ,t,x)),\ h(\zeta ')=(\zeta ',1),\ g(\zeta )= (1-\zeta , \zeta ). \end{aligned}$$

To shorten the notation, we assume that the Fenchel conjugate of $W(\omega ,x, \cdot )$ satisfies (GD) so that $W^{hom}$ is Gâteaux differentiable.

• (a) The case of random single delay which depends on the space variable. Concerning the distributed delays, we first consider the family of random measures associated with the random single delay of the first structure described in Sect. . These measures models a disease whose incubation period varies according to the spatial heterogeneities: we have ${\mathbf {m}}_t^\varepsilon (\omega )= \frac{1}{\#\tau }\delta _{t-\tau (\omega ,\frac{\cdot }{\varepsilon })}$, and the reaction functional is given by

  $$\begin{aligned} a\left( \omega , t,\frac{\cdot }{\varepsilon }\right) ( 1-u_\varepsilon (\omega ,t)) \frac{1}{\#\tau }u_\varepsilon \left( \omega , t- \tau \left( \omega ,\frac{\cdot }{\varepsilon }\right) \right) - b\left( \omega , t,\frac{\cdot }{\varepsilon }\right) u_\varepsilon (\omega ,t). \end{aligned}$$

Then, assuming that for $\mathbf {P}$ a.s. $\omega \in \Sigma$ it hold $\sup _\varepsilon \Phi _\varepsilon (\eta _\varepsilon (0))<+\infty$ and that $\eta _\varepsilon$ converges to $\eta$ in $C_c((-\infty , 0], L^2(\Omega ))$ whose support is included in $[-M, 0]$ for some $M>0$, according to Theorem 5.1, we can say that when $\varepsilon$ is very small compared to the size of the domain, a deterministic model, well aware with the evolution of the infected population, is given by the homogenized problem

$$\begin{aligned} ({\mathcal P}^{hom}) \left\{ \begin{array}{l} \displaystyle \frac{d u}{ dt}(t) -\mathrm{div}D_\xi W^{hom}(\nabla u(t)) = F^{hom}(t, u(t)) \hbox { for a.e. } t\in (0,T)\\ u(t)= \eta (t), \quad 0\le \eta (t)\le \overline{\rho } \hbox { for all } t\in (-\infty , 0],\\ 0\le u(t)\le \overline{\rho } \hbox { for all } t\in (-\infty , 0],\\ u(t)\in H^1(\Omega ),\ \mathrm{div}D_\xi W^{hom}(\nabla u(t))\in L^2(\Omega )\hbox { for all } t\in [0,T], \\ u(t)=0\hbox { on }\partial \Omega \hbox { for all } t\in [0,T], \end{array} \right. \end{aligned}$$

where,

$$\begin{aligned} F^{hom}(t, u(t))= & {} \frac{1}{\#\tau } \mathbf {E}\left[ \int _Y a(\cdot ,t,y) u(t-\tau (\cdot ,y))dy\right] ( 1-u(t)) \\&-\,\mathbf {E}\left[ \int _Y b(\cdot ,t,y)\ dy\right] u(t). \end{aligned}$$

We see that the growth rate of the uninfected population and the time delay have been mixed and averaged. Let us look at two types of spatial environments.

The$\varepsilon$-random checkerboard-like environment. Assume that the spread of the disease occurs in a $\varepsilon$-random checkerboard-like environment modeling a mosaic of two kinds of small habitats, with two virus strains: the growth rate of the uninfected and infected population, the diffusion density, together with the time delay (incubation period), take two values at random on the lattice spanned by the cell $Y=(0,1)^N$ at scale 1, say $a^-, a^+$, $b^-, b^+$, $W^-, W^+$, and $\tau ^-, \tau ^+$ respectively, with a probability $p>0$ and $1-p$ in each cell. The $\varepsilon$-random checkerboard-like environment is then modeled by a ergodic dynamical system $(\Sigma , {\mathcal A}, \mathbf {P}_p, (T_z)_{z\in \mathbb {Z}^N})$ where $(\Sigma , {\mathcal A}, \mathbf {P}_p)$ is a Bernoulli product probability space, and $T_z$ the shift operator. In this specific case we have

$$\begin{aligned} F^{hom}(t, u(t))= & {} \frac{1}{2}\left[ p a^-u(t-\tau ^-)+ (1-p)a^+u(t-\tau ^+)\right] (1-u(t) )\\&- \left[ p b^-+(1-p)b^+\right] u(t). \end{aligned}$$

The Poisson point process environment We now consider the Poisson point environment with $N=2$, a little more realistic, where spherical heterogeneities with radius $\varepsilon$, are independently randomly distributed with a given frequency $\lambda$ following a Poisson point process with intensity $\lambda$. Recall that this random environment is modeled by an ergodic dynamical system $(\Sigma , {\mathcal A}, \mathbf {P}_\lambda , (T_z)_{z\in \mathbb {R}^2})$ where $T_z\omega =\omega -z$,⁸ and, for every bounded Borel set B, and every $k\in \mathbb {N}$,

$$\begin{aligned} \mathbf {P}_\lambda (\#(\Sigma \cap B)=k)=\lambda ^k{\mathcal L}_2 (B)^k\; \frac{\exp (-\lambda {\mathcal L}_2(B))}{ k!} \end{aligned}$$

so that $\mathbf {E}_\lambda \left[ \#\Sigma \cap B\right] =\lambda {\mathcal L}_2(B)$. Assume that the disease consists of two virus strains, one of these being concentrated in the environment, at scale 1, made up of all the balls with radius R centered at the points of the Poisson process. Then, given $R>0$, we define the random density W associated with the random diffusion part, by

$$\begin{aligned} W(\omega , x, \xi )= {\left\{ \begin{array}{ll} \displaystyle W^- (\xi ) \hbox { if } x\in \bigcup _{i\in \mathbb {N}}B_R(\omega _i),\\ W^+(\xi )\ \hbox {otherwise.} \end{array}\right. } \end{aligned}$$

Similarly for the reaction functional, we set

$$\begin{aligned} a(\omega ,t, x)={\left\{ \begin{array}{ll} \displaystyle a^- (t) \hbox { if } x\in \bigcup _{i\in \mathbb {N}}B_R(\omega _i),\\ a^+(t)\ \hbox {otherwise,} \end{array}\right. } b(\omega ,t, x)={\left\{ \begin{array}{ll} \displaystyle b^- (t) \hbox { if } x\in \bigcup _{i\in \mathbb {N}}B_R(\omega _i),\\ b^+(t)\ \hbox {otherwise,} \end{array}\right. } \end{aligned}$$

and, for the time delays, $\tau (\omega ,t, x)={\left\{ \begin{array}{ll} \displaystyle \tau ^- \hbox { if } x\in \bigcup _{i\in \mathbb {N}}B_R(\omega _i),\\ \tau ^+\ \hbox {otherwise.} \end{array}\right. }$

Noticing that $(\exists \omega \in \Sigma ,\quad y\in \bigcup _{i\in \mathbb {N}} B_R(\omega _i)) \iff \# ( \Sigma \cap B_R(y) ) \ge 1,$ and using Fubini’s theorem, we can express the homogenized reaction functional by

$$\begin{aligned}&F^{hom}(t, u(t))=\frac{1}{\#\tau } \mathbf {E}_\lambda \left[ \int _Y a(\cdot ,t,y) u (t-\tau (\cdot ,y)) dy\right] ( 1-u(t) )\\&-\mathbf {E}_\lambda \left[ \int _Y b(\cdot ,t,y)\ dy\right] u(t). \end{aligned}$$

Indeed

$$\begin{aligned}&\mathbf {E}_\lambda \left[ \int _Y a(\cdot ,t,y) u (t-\tau (\cdot ,y) ) dy\right] (1-u(t)) \\&\quad =a^+(t) u(t-\tau ^+)( 1-u(t) ) \int _\Sigma \int _{(0,1)^2} \mathbb {1}_{\left[ \# (\Sigma \cap B_R(y)) =0\right] }(\omega , y) dy\ d\mathbf {P}(\omega )\\&\qquad +\,a^-(t) u(t-\tau ^-)( 1-u(t) ) \int _\Sigma \int _{(0,1)^2} \mathbb {1}_{\left[ \# (\Sigma \cap B_R(y) ) \ge 1\right] }(\omega , y) dy\ d\mathbf {P}(\omega )\\&\quad =a^+(t) u(t-\tau ^+)( 1-u(t) ) \int _{(0,1)^2}\int _\Sigma \mathbb {1}_{\left[ \# (\Sigma \cap B_R(y) ) =0\right] }(\omega , y) d\mathbf {P}(\omega )\ dy\\&\qquad +\,a^-(t) u{(t-\tau ^-)( 1-u(t))} \int _{(0,1)^2}\int _\Sigma \mathbb {1}_{\left[ \# (\Sigma \cap B_R(y)) \ge 1\right] }(\omega , y) d\mathbf {P}(\omega )\ dy\\&\quad =a^+(t) u(t-\tau ^+)( 1-u(t) ) \exp (-\lambda \pi R^2)+ a^-(t) u(t-\tau ^-)( 1-u(t) ) {(1- \exp (-\lambda \pi R^2))} \\&\quad =\left[ a^-(t) u(t-\tau ^-) + {(a^+(t) u(t-\tau ^+)-a^-(t) u(t-\tau ^-))} \exp (-\lambda \pi R^2) \right] (1-u(t)) \end{aligned}$$

and, from a similar calculation

$$\begin{aligned} \mathbf {E}_\lambda \left[ \int _Y \beta (\cdot ,t,y)\ dy\right] u(t) =\left[ b^-(t)+ ( b^+(t)-b^-(t) ) \exp (-\lambda \pi R^2) \right] u(t). \end{aligned}$$

Consequently, the homogenized reaction functional is given by

$$\begin{aligned} F^{hom}(t, u(t))&=\frac{1}{2}\left[ a^-(t) u(t-\tau ^-) + ( a^+(t) u(t-\tau ^+)-a^-(t) u(t-\tau ^-) ) \right. \\&\qquad \qquad \left. \exp (-\lambda \pi R^2) \right] ( 1-u(t) ) \\&\quad +\,\frac{1}{2}\left[ b^-(t)+ ( b^+(t)-\beta ^-(t) ) \exp (-\lambda \pi R^2) \right] u(t).\ \end{aligned}$$

• (b) The case of a random multiple delays. We consider now the case when the family of random measures corresponds to the second structure of Sect. 5.2.2 derived from Example 2.1, that is ${\mathbf {m}}_t^\varepsilon (\omega )= \sum _ {i\in \mathbb {N}}\mathbf {d}_k(\omega , \frac{\cdot }{\varepsilon }) \delta _{t-\tau _k}$. Then, under the same hypotheses, the limit second member becomes

  $$\begin{aligned} F^{hom}(t, u(t))= & {} ( 1-u(t)) \sum _{k\in \mathbb {N}} \mathbf {E}\left[ \int _Y a(\cdot ,t,y) \mathbf {d}_k(\cdot ,y)\ dy\right] u(t-\tau _k) \\&-\,u(t) \mathbf {E}\left[ \int _Y b(\cdot ,t,y)\ dy\right] \end{aligned}$$

which can be expressed in the two previous probabilistic cases by reproducing the calculations of a). We see that the growth rate of the uninfected population and the time delays coefficients have been mixed and averaged in such a way that the limit delays reaction-diffusion problem is a reaction diffusion equation with a multiple delays associated with the measure $\sum _{k\in \mathbb {N}} \mathbf {E}\left[ \int _Y a(\cdot ,y) \mathbf {d}_k(\cdot ,y)\ dy\right] \delta _{t-\tau _k}$ incorporating the growth rate.

Appendix B. Vector measures

We give a brief summary on the concept of vector measure used to define the time-delays operators in Sect. 2.

Definition B.1

Let $\mathbb {T}$ be a locally compact space and denote by ${\mathcal B}(\mathbb {T})$ its Borel $\sigma$-algebra. Given a Banach space ${\mathcal Y}$, we call ${\mathcal Y}$-valued Borel vector measure, any countably additive set function $\mathbf {m}: {\mathcal B}(\mathbb {T})\rightarrow \mathcal { Y}$.

Let X be another Banach space. We denote by ${\mathcal {E}}(\mathbb {T}, X)$ the space of step functions from $\mathbb {T}$ into X, i.e. the space of functions $S: \mathbb {T}\rightarrow X$ of the form $S=\sum _{i\in I} \mathbb {1}_{B_i} S_i$ where I is any finite set, $B_i\in {\mathcal B}(\mathbb {T})$, and $S_i\in X$. Assume that there exists a continuous bilinear mapping $\mathbf {B}: {\mathcal Y}\times X\ni (u,v)\mapsto \mathbf {B}( u, v)\in X$, it follows that we can integrate with respect to $\mathbf {m}$, step functions $S\in {\mathcal {E}}(\mathbb {T}, X)$, and define an element of X, according to the formula

$$\begin{aligned} \int S d \mathbf {m}:= \sum _{i\in I} \mathbf {B}( \mathbf {m}(B_i), S_i)\in X. \end{aligned}$$

The definition of $\int S d \mathbf {m}$ depends only on S and is independent of the particular way in which S is written (see [13, Chapter II, 7, 1, Proposition 1]). We set

$$\begin{aligned} {\mathcal B}(\mathbb {T})\ni B\mapsto \mu (B)= & {} \sup \left\{ \sum _{i\in I} \Vert \mathbf {m}(B_i)\Vert _{\mathcal Y}: I \text{ finite, } \{B_i\}_{i\in I}\right. \\&\qquad \qquad \left. \subset {\mathcal B}(\mathbb {T}) \text{ pairwise } \text{ disjoint, } B=\cup _{i\in I} B_i\right\} . \end{aligned}$$

Definition B.2

The set function $\mu : {\mathcal B}(\mathbb {T})\rightarrow [0, +\infty ]$ is called the variation of $\mathbf {m}$ and is denoted by $\Vert \mathbf {m}\Vert$. The vector measure $\mathbf {m}$ has finite variation if $\Vert \mathbf {m}\Vert (\mathbb {T})<+\infty$.

The following proposition is an easy consequence of the definitions above.

Proposition B.1

Let $\mathbf {m}$ be a ${\mathcal Y}$-valued Borel measure with finite variation. Then $\Vert \mathbf {m}\Vert$ is a (scalar) nonnegative Borel measure and, for all $S\in {\mathcal {E}}(\mathbb {T}, X)$, we have

$$\begin{aligned} \left\| \int S d\mathbf {m}\right\| _X\le C_\mathbf {B}\sup _{t\in \mathbb {T}} \Vert S(t)\Vert _X \Vert \mathbf {m}\Vert (\mathbb {T}) \end{aligned}$$

(B.1)

where $C_{\mathbf {B}}$ is the norm of the bilinear mapping $\mathbf {B}$.

Let denote by ${\mathcal {E}^\infty }(\mathbb {T}, X)$ the space of functions from T into X which are uniform limit of step functions. Then we have

Proposition B.2

(Extension of the integral). Let $\mathbf {m}$ be a ${\mathcal Y}$-valued Borel measure with finite variation. Then, the integral $\int S d \mathbf {m}$, defined for all $S\in {\mathcal {E}}(\mathbb {T}, X)$, can be extended in a standard way to the space ${\mathcal {E}^\infty }(\mathbb {T}, X)$, and for all $u\in {\mathcal {E}^\infty }(\mathbb {T}, X)$, we have

$$\begin{aligned} \left\| \int u d\mathbf {m}\right\| _{X} \le \int \Vert u\Vert _{X} d \Vert \mathbf {m}\Vert . \end{aligned}$$

(B.2)

Sketch of the proof

First recall the standard extension process. Let $u\in {\mathcal {E}^\infty }(\mathbb {T}, X)$, $u=\lim _{n\rightarrow +\infty } S_n$ for the uniform norm, where $S_n\in {\mathcal {E}}(\mathbb {T}, X)$ for all $n\in \mathbb {N}$. Since $(S_n)_{n\in \mathbb {N}}$ is a Cauchy sequence for the uniform norm, from (B.1), $(\int S_n d\mathbf {m})_{n\in \mathbb {N}}$ is a Cauchy sequence in X, thus converges to some element U in X. Let us show that U does not depend on the choice of $(S_n)_{n\in \mathbb {N}}$. Let $(T_n)_{n\in \mathbb {N}}$ be any sequence in ${\mathcal {E}}(\mathbb {T}, X)$ uniformly converging to u. Then, from (B.1)

$$\begin{aligned} \left\| \int (T_n-S_n) d\mathbf {m}\right\| _X\le \sup _{t\in \mathbb {T}} \Vert (T_n- S_n)(t)\Vert _X \Vert \mathbf {m}\Vert (\mathbb {T}), \end{aligned}$$

so that $\lim _{n\rightarrow +\infty }\int T_n d\mathbf {m}= \lim _{n\rightarrow +\infty } \int S_n d\mathbf {m}= U$. Inequality (B.2) is easy to establish for functions in ${\mathcal {E}^\infty }(\mathbb {T}, X)$ (see [13, Chapter II, 7, 2, Proposition 2]). For the general case, argue by density. $\square$

Proposition B.2 justifies the notation $\int u d\mathbf {m}$ for all $u\in {\mathcal {E}^\infty }(\mathbb {T}, X)$. It is well known that ${\mathcal {E}^\infty }(\mathbb {T}, X)$ contains the space $C_c(\mathbb {T}, X)$ of continuous functions from $\mathbb {T}$ into X with compact support (see [13, Chapter III, 14, 5, Corollary 2]). Therefore, $\int u\ d \mathbf {m}$, which is sometimes written $\int _{\mathbb {T}} u(t) d\mathbf {m}(t)$, has a meaning for any $u\in C_c(\mathbb {T}, X)$, and defines an element of X. Note also that for $B\in {\mathcal B}(\mathbb {T})$, and $u\in C_c(\mathbb {T}, X)$, $\mathbb {1}_B u$ is uniform limit of step functions so that $\int _{\mathbb {T}} \mathbb {1}_B u\ d\mathbf {m}$, written $\int _B u d\mathbf {m}$, has a meaning and defines an element of X. We refer the reader to [13] for a comprehensive review on vector measures.

Appendix C. Notion of CP-structured reaction functionals

The definition and the theorem below are extracted from [2].

Definition C.1

Let denote by $\mathbb {R}^\Omega$ the set of all the maps from $\Omega$ into $\mathbb {R}$. A map $F: [0,+\infty )\times L^2(\Omega )\rightarrow \mathbb {R}^\Omega$ is called a CP-structured reaction functional, if there exists a Borel measurable function $f: [0, +\infty )\times \mathbb {R}^N\times \mathbb {R}\rightarrow \mathbb {R}$ such that for all $t\in [0, +\infty )$ and all $v\in L^2(\Omega )$, $F(t, v)(x)= f(t,x, v(x))$, which fulfills the following structure conditions:

$$\begin{aligned} f(t,x,\zeta )= r(t,x)\cdot g(\zeta )+ q(t,x) \end{aligned}$$

with

• $g :\mathbb {R}\rightarrow \mathbb {R}^l$ is a locally Lipschitz continuous function;

• for all $T>0$, r belongs to $L^\infty ([0,T]\times \mathbb {R}^N,\mathbb {R}^l)$;

• for all $T>0$, q belongs to $L^2(0,T, L^2_\mathrm{loc}(\mathbb {R}^N))$.

Furthermore f must satisfy the following condition:

• (CP) there exist a pair $(\underline{f}, \overline{f})$ of functions $\underline{f}, \overline{f} : [0, +\infty )\times \mathbb {R}\rightarrow \mathbb {R}$ with $\underline{f} \le 0\le \overline{f}$ and a pair $(\underline{\rho }, \overline{\rho })$ in $\mathbb {R}^2$ with $\underline{\rho }\le \overline{\rho }$, such that each of the two following ordinary differential equations

  $$\begin{aligned} \underline{\text{ ODE }} \left\{ \begin{array}{l} \underline{y}'(t)= \underline{f}(t, \underline{y}(t) )\hbox { for a.e. }t\in [0, +\infty )\\ \underline{y}(\text{0 })=\underline{\rho } \end{array} \right. \quad \overline{\text{ ODE }} \left\{ \begin{array}{l} \overline{y}'(t)=\overline{f}(t, \overline{y}(t)) \hbox { for a.e. }t\in [0, +\infty )\\ \overline{y}(\text{0 })=\overline{\rho } \end{array} \right. \end{aligned}$$

  admits at least a solution denoted by $\underline{y}$ for $\underline{\text{ O } \text{ DE }}$ and by $\overline{y}$ for $\overline{\text{ O } \text{ DE }}$ satisfying for a.e. $(t,x)\in (0, +\infty )\times \mathbb {R}$

  $$\begin{aligned} \underline{f}(t,\underline{y}(t))\le f(t,x, \underline{y}(t)) \quad \text{ and } \quad f(t,x, \overline{y}(t))\le \overline{f}(t, \overline{y}(t)). \end{aligned}$$

The map F is referred to as a CP-structured reaction functional associated with (r, g, q), and f as a CP-structured reaction function associated with (r, g, q). The map F is referred to as a regular CP-structured reaction functional and f as a regular CP-structured reaction function if furthermore, for all $T>0$, $r\in W^{1,1}(0,T, L^2_\mathrm{loc}(\mathbb {R}^N, \mathbb {R}^l))$ and $q\in W^{1,1}(0,T, L^2_\mathrm{loc}(\mathbb {R}^N))$.

Theorem C.1

Let F be a CP-structured reaction functional, with $\underline{\rho }$, $\overline{\rho }$ and $\underline{y}$, $\overline{y}$ given by (CP), and let $\Phi$ be a functional of the calculus of variations (3.1). Assume that $a_0\underline{\rho }\le h\le a_0\overline{\rho }$ on $\partial \Omega$. Then for any $T>0$, the Cauchy problem

$$\begin{aligned} ({\mathcal P}) \left\{ \begin{array}{l} \displaystyle \frac{du}{dt}(t)+ D\Phi (u(t))= F(t, u(t))\hbox { for a.e. } t\in (0, T)\\ u(0)= u_0,\quad \underline{\rho }\le u_0\le \overline{\rho },\quad u_0\in \overline{\mathrm{dom}(D\Phi )} \end{array} \right. \end{aligned}$$

admits a unique solution $u\in C([0,T], L^2(\Omega ))$ satisfying the following bounds in [0, T]: $\underline{y}(T)\le \underline{y}(t)\le u(t)\le \overline{y}(t)\le \overline{y}(T)$. If furthermore F is a regular CP-structured reaction functional, then u admits a right derivative for all $t\in [0, T[$ which satisfies $({\mathcal P})$ for all $t\in [0, T[$.

Appendix D. Estimate of the right derivative

Lemma D.1

Let X be a Hilbert space, $T>0$, $G\in W^{1,1}(0,T, X)$ and $\Phi : X\rightarrow \mathbb {R}\cup \{+\infty \}$ be a convex proper lower semicontinuous functional. Let u satisfy

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{du}{dt}(t)+ \partial \Phi (u(t))\ni G(t)\hbox { for a.e. }t\in (0,T),\\ u(0)\in \overline{\mathrm{dom}(\partial \Phi )}. \end{array} \right. \end{aligned}$$

(D.1)

Then the right derivative of u satisfies for all $t\in ]0, T[$ the following estimate

$$\begin{aligned} \left\| \frac{d^+u}{{dt}}(t)\right\| _X\le \frac{1}{t} \int _0^t \left\| \frac{du}{dt}(s)\right\| _X ds + \int _0^t \left\| \frac{ dG}{dt}(s)\right\| _X ds. \end{aligned}$$

For a proof consult [2, Lemma 3.3]