1 Introduction
King et al. [Reference King, Brown, Sargin, Bratlie and Beckman5, Reference King, Brown, Sargin, Bratlie and Beckman6] proposed models that describe reaction–diffusion of oxygen through a protective shell encapsulating a core of donor cells to determine conditions so that hypoxia of the donor cells can be avoided. This geometry introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. The results of King et al. are restricted to numerical computation of stationary solutions assuming spherical geometries. Their results were made rigorous in [Reference de Jong, Sterk, Abramian, Andrianov and Gaiko2]. In [Reference de Jong, Prokert and Sterk1], the corresponding parabolic partial differential equation (PDE) is studied for general core–shell geometries. It is shown that the PDE is well posed and that stationary solutions are stable. These last results crucially depend on the monotonicity of the oxygen consumption and are derived from Michaelis–Menten kinetics. However, during critical cell states such as partial death of donor cells, these monotonicity conditions will not be satisfied. In this paper, we consider the PDE for general consumption, that is, consumption is bounded, nonnegative and zero for negative concentrations.
In the classical theory on existence and uniqueness for nonlinear reaction–diffusion equations, it is typically assumed that the reaction term has asymptotics similar to an odd degree polynomial [Reference Marion8, Reference Robinson9]. This means that the reaction term is unbounded. Consequently, the regularity of the constructed solution will depend on the degree of the leading asymptotics of the reaction term. However, in our setting, the reaction term is bounded which leads to a degenerate setting with respect to classical theory.
We construct the solutions using the Galerkin method (see [Reference Marion8, Reference Robinson9]). Although the nonlinearity in our setting does not exactly satisfy the classical results in [Reference Marion8, Reference Robinson9], a Galerkin set-up still works. Additionally, we are dealing with a so-called diffraction problem [Reference Ladyzhenskaya7] meaning that the diffusion coefficient is discontinuous. However, it turns out that the bounds on the term with the Laplacian guarantee that the bounds for the Galerkin approximation are not in danger. Finally, as with diffraction problems, the loss of regularity resulting from the discontinuity will not be visible when considering the well-posedness of weak solutions, but only when we consider the well-posedness of strong solutions.
This paper is organised as follows. In Section 2, we present the statement of the problem and in Section 3, we provide preliminaries. In Section 4, we first establish the global existence and uniqueness of the weak solutions in Section 4.1, followed by the main results on global existence and uniqueness of the strong solutions in Section 4.2. Finally, in Section 5, we provide conclusions and some remarks for future work.
2 Statement of the problem
We start with a description of the core–shell geometry. For an integer
$N \ge 2$
, let
$\Omega \subset \mathbb {R}^N$
with
$\overline {\Omega }$
compact, 
be the boundary of
$\Omega $
,
$\nu \colon S \to \mathbb {R}^N$
be the outward unit vector and
$T> 0$
be a constant. Let
$\Gamma ~(\subset \Omega )$
be an
$(N-1)$
-dimensional surface that divides
$\Omega $
into two open domains
$\Omega _1$
and
$\Omega _2$
, that is,
$\Gamma = \overline {\Omega }_1\cap \overline {\Omega }_2$
,
$\Omega = \Omega _1 \cup \Omega _2 \cup \Gamma $
. We suppose
$\partial \Omega _1 = \Gamma $
and
$\partial \Omega _2 = S \cup \Gamma $
(see Figure 1). We take
$S,\Gamma $
of class
$C^2$
.
The governing equations of our problem are given by
$$ \begin{align} \frac{d u}{d t} - b \Delta u &=f(u) \quad\text{in}~\Omega_i \times (0, T),\ i=1, 2, \end{align} $$
$$ \begin{align} u &= 0 \quad \text{on}~S \times (0, T), \end{align} $$
$$ \begin{align} [u]_\Gamma &= 0 \quad \text{on}~\Gamma \times (0, T), \end{align} $$
$$ \begin{align} [ b \nabla u \cdot \nu ]_{\Gamma} &=0 \quad \text{on}~\Gamma \times (0, T), \end{align} $$
$$ \begin{align} u &= u_0 \quad \text{in}~\Omega, \text{at} t=0, \end{align} $$
where
$b: \overline {\Omega }_1 \cup \Omega _2 \rightarrow \mathbb {R} $
is given by
with constants
$b_1, b_2>0$
. Here, the discontinuous diffusion term,
$b \Delta u$
, is called the diffraction Laplacian,
$u_0\colon \Omega \to \mathbb {R}$
is a given initial value and
$[\cdot ]_{\Gamma }\colon \Gamma \times (0,T)\rightarrow \mathbb {R}$
denotes the difference of limiting values on
$\Gamma $
, which is defined by
where
$u_1$
and
$u_2$
are the restrictions of u to
$\Omega _1$
and
$\Omega _2$
, respectively. Finally, we consider a linear operator
$f: L^2(\Omega ) \rightarrow L^2(\Omega )$
that is assumed to be Lipschitz and satisfies the conditions:
$$ \begin{align} \nonumber &(u, f(u) ) \leq K \quad\text{for all } u \in L^2(\Omega), \\ &\|f\|_{ L^2(\Omega) }\leq K, \; \end{align} $$
with
$K>0$
. Here,
$(\cdot \,, \cdot )$
denotes the inner product on
$L^2(\Omega )$
and
$\|f\|_{L^2(\Omega )}$
is the operator norm given by
Equations (2.1)–(2.5) are for the transformed concentration. The concentration can be retrieved by
$v= c_0 - u$
with
$v=c_0$
on
$S \times (0,T)$
(see [Reference de Jong, Prokert and Sterk1, Appendix A]). We assume that the consumption
$g(v):=f(c_0-v)$
is bounded, nonnegative and zero for negative concentrations. Then, (2.6) is satisfied.
Core–shell geometry of an encapsulated cell.
3 Preliminaries
3.1 Notation
We define 
, 
and
$V^{*}$
,
$H^{*}$
as their dual spaces, respectively. The inner product on V is defined by
$(u,v)_V = (u,v)_H + (\nabla u, \nabla v)_H$
. We denote by
$(\cdot \,, \cdot )$
the inner product on H, and by
$\langle \cdot \,,\cdot \rangle $
the pairing between
$V^{*}$
and V. Then,
$V \subset \subset H=H^* \subset V^*$
, where we write
$V\subset \subset H$
to emphasise the compactness of the embedding of V in H. Let
$X=V,H$
or
$V^*$
. The
$L^p(0,T;X)$
-norm
$(p=2,\infty )$
is
We define
$b_{\max }:=\max \{b_1,b_2\}$
and
$b_{\min }:=\min \{b_1,b_2\}$
. We denote
$u_n$
converging weakly to u in X by
$u_n \rightharpoonup u$
in X. We reserve
$c>0$
to denote generic positive constants that do not depend on the relevant parameters and variables.
3.2 Diffraction Laplacian
We introduce the bilinear form
$a: V \times V \rightarrow \mathbb {R}$
given by
$$ \begin{align*} a(u,v) = \int_\Omega b \nabla u \cdot \nabla v \,dx. \end{align*} $$
This induces a linear operator
$A: V \rightarrow V^*$
given by
$ \langle Au, v \rangle = a(u,v)$
for
$v \in V$
.
Note that
$a(\cdot \,,\cdot )$
is bounded (
$|a(u,v)| \leq c \Vert u \Vert _V \Vert v \Vert _V$
) and coercive (
$c \Vert u \Vert _V^2 \leq a(u,u)$
). Consequently, by the Lax–Milgram theorem, A is bijective. Also, observe that
$A^{-1}$
is bounded since for
$Au = f \in V^*$
, we can write
$ \Vert u \Vert ^2_V \leq c a(u,u) = c \langle f, u \rangle \leq c \Vert f \Vert _{V^*} \Vert u \Vert _V,$
which gives
$\Vert u \Vert _V \leq c \Vert f \Vert _{V^{*}}.$
Define
$S: H \rightarrow H$
by
$S =r \circ A^{-1} \circ \iota $
, with
$A^{-1}:V^{*}\rightarrow V$
,
$\iota $
the inclusion map
$\iota \colon H\rightarrow V^{*}$
and r the restriction map
$r\colon V\rightarrow H$
. Since A is bijective and
$A^{-1}$
bounded, S is compact. Abusing notation, we write
$A = S^{-1}$
and consider
$A\colon H\rightarrow H$
. Observe that A is symmetric. From spectral theory for unbounded operators, A can be represented by
$Au=\sum _{j=1}^{\infty }\lambda _{j}(u,w_j)w_j,$
where
$\lambda _j$
and
$w_j$
are the real eigenvalues and eigenfunctions of A, respectively. From the smoothness on the boundaries, the domain of A is given by
$D(A) = \{ u \in V : u|_{\Omega _i} \in H^2(\Omega _i), \ u \ \mathrm{satisfies} \ (2.4) \}.$
The inner product on
$D(A)$
is given by
$(u,v)_{D(A)} = (Au,Av)$
.
3.3 Projections
We define the projection
$P_n$
which maps
$u\in H$
into the first n eigenfunctions of A:
$P_n u := \sum _{j=1}^{n} (u,w_j)w_j$
. The projection orthogonal to
$P_n$
is defined by
$Q_n:= \mathrm{id} - P_n$
.
3.4 Classical results
We review the classical results from [Reference Robinson9]. Most of these results have been reduced to fit the application. A page number is included so that the full statement can be recovered.
Lemma 3.1 [Reference Robinson9, page 199].
If
$X=H,V$
or
$V^*$
, then
$$ \begin{align*} \Vert P_n u \Vert_X \leq \Vert u \Vert_X , \quad P_n u \rightarrow u {\ \mathrm{in} \ } X \quad (n\rightarrow \infty). \end{align*} $$
Lemma 3.2 [Reference Robinson9, page 218].
Let
$\mathcal {O}$
be a bounded open set in
$\mathbb {R}^m$
and let
$g_j$
be a sequence of functions in
$L^2(\mathcal {O})$
with
$\Vert g_j \Vert _{L^p(\mathcal {O})} \leq C$
. If
$g \in L^2(\mathcal {O})$
and
$g_j \rightarrow g$
pointwise almost everywhere, then
$g_j \rightharpoonup g$
in
$L^2(\mathcal {O})$
.
Theorem 3.3 [Reference Robinson9, page 191].
Suppose that
$$ \begin{align*} u \in L^2(0,T;V)\quad \textit{and}\quad \frac{du}{dt} \in L^2(0,T;V^*). \end{align*} $$
Then
$u \in C^0([0,T], H)$
, with the caveat that u may have to be adjusted on a set of measure zero.
4 Well-posedness results
We will start with the well-posedness of the weak solutions which we then straightforwardly extend to the well-posedness of the strong solution.
4.1 Weak solutions
We consider
$$ \begin{align} \frac{du}{dt} + Au = f(u), \end{align} $$
as an equality in
$L^2(0,T;V^*)$
.
Theorem 4.1 (Well-posedness of weak solutions).
For any
$T>0$
, (4.1) with
$u(0)=u_0 \in H$
has a unique weak solution u such that
$u, {du}/{dt} \in L^2(0,T;V^*)$
and
$u \in C^0([0,T];H) $
with the caveat that it may have to be adjusted on a set of measure zero. Furthermore,
$u_0 \mapsto u(t)$
is in
$C^0(H;H)$
.
Proof. We consider the solutions expressed by the first n eigenfunctions of A:
$$ \begin{align*} u_n(t) = \sum_{j=1}^{n} u_{nj}(t) w_j, \end{align*} $$
satisfying
$$ \begin{align*} \bigg( \frac{du_n}{dt} , w_i \bigg) + (A u_n , w_i ) = (f(u_n), w_i ), \quad 1 \leq i \leq n, \end{align*} $$
with
$(u_n(0),w_i) = (u_0,w_i)$
. Define
$H_n := P_n H \subset H$
. We need to solve the initial value problem
$$ \begin{align} \frac{dv}{dt} + Av = P_n f(v), \quad v(0) = P_n u(0), \end{align} $$
on the finite-dimensional space
$H_n$
. The mapping
$v \mapsto -Av + P_n f(v)$
is Lipschitz continuous from
$H_n$
to
$H_n$
. By standard existence–uniqueness results for ordinary differential equations (ODEs), (4.2) has a unique solution on some finite interval
$[0,T]$
with T dependent on n and
$u_0$
. We will see that the solution exists for all
$T>0$
.
Consider the inner product of (4.2) with
$u_n$
:
$$ \begin{align*} \bigg(\frac{du_{n}}{dt},u_n \bigg) + (A u_{n},u_n) &= (P_n f(u_n),u_n). \end{align*} $$
Observe that
$(P_n f(u_n),u_n) = (f(u_n), P_n u_n) = (f(u_n),u_n)$
. From the assumption that
$(u,f(u))\leq K$
and the coercivity of
$a(\cdot \,,\cdot )$
,
$$ \begin{align*} \frac12 \frac{d \|u_n\|^2_H}{dt} + b_{\min} \Vert u_n \Vert_V^2 \leq K. \end{align*} $$
Integrating both sides over t between
$0$
and T gives
$$ \begin{align*} \frac12 \|u_n(T)\|^2_H + b_{\min} \int_{0}^T \Vert u_n \Vert_V^2 \,dt \leq K T + \frac1{2} \|u(0)\|^2_H. \end{align*} $$
We define
$\gamma := K T + \frac 1{2} \|u(0)\|^2_H $
. Then, we obtain the bounds:
$$ \begin{align} \sup_{t \in [0,T]} \|u_n(t)\|^2_H & \leq 2 \gamma, \end{align} $$
$$ \begin{align} \int_{0}^T \Vert u_n \Vert_V^2 \,dt &\leq \frac{\gamma}{b_{\min}}. \end{align} $$
Observe that
$\gamma $
is linear in T. Hence, (4.3) with local existence of solutions for (4.2) gives existence of solutions for any
$T>0$
. From (4.3) and (4.4),
$u_n$
is uniformly bounded in
$L^\infty (0,T; H)$
and
$L^2(0,T; V)$
.
Since
$\|f\|_H \leq K$
, we see that
$f(u_n)$
is uniformly bounded in
$L^2(0,T; H)$
and
$Au_n$
is uniformly bounded in
$L^2(0,T; V^*)$
. Hence, from (4.2),
$du_n/dt$
is uniformly bounded in
$L^2(0,T; V^*)$
. By Aloaglu’s compactness theorem, we can extract a weakly convergent subsequence
$u_n$
, with
$u_n \rightharpoonup u$
in
$L^2(0,T;V)$
and
$f(u_n) \rightharpoonup \chi $
in
$L^2(0,T;H)$
. The strong convergence
$u_n \rightarrow u $
in
$L^2(0,T;H)$
is obtained by using Lemma 3.2.
Now we want to show that
$P_n f(u_n) \rightharpoonup \chi $
in
$L^2(0,T;H)$
. We have
$$ \begin{align} \int_{\Omega_T}(P_n f(u_n) - \chi) \phi \,dx\,dt &= \int_{\Omega_T} (f(u_n) - \chi) \phi \,dx\,dt - \int_{\Omega_T} Q_n f(u_n)\phi \,dx\,dt \end{align} $$
for all
$\phi \in L^2(0,T;H)$
. Recall that
$f(u_n) \rightharpoonup \chi $
in
$L^2(0,T;H)$
. So we just need to consider the
$Q_n$
term in (4.5). Observe that
$\| Q_n f(u_n) \|_H = \| f(u_n) \|_H \leq K $
. We can consider
$\phi = \sum _{j=1}^m \alpha _j(t) \phi _j$
, where
$\alpha _j \in L^2(0,T)$
and
$ \phi _j \in C^\infty _c(\Omega ))$
since
$\phi $
is dense in
$ L^2(0,T;H)$
. From Lemma 3.1,
$Q_n \phi _j \rightarrow 0$
in H and we have shown that
$P_n f(u_n) \rightharpoonup \chi $
in
$L^2(0,T;H)$
. By combining the results, we arrive at the equality
$$ \begin{align*} \frac{du}{dt} + Au = \chi, \end{align*} $$
which holds in the dual space
$L^2((0,T); V^*)$
.
Next, we show that
$\chi = f(u)$
. Since
$u_n \rightarrow u$
in
$L^2(0,T;H)$
, there exists a subsequence
$u_{n_j}$
such that
$u_{n_j}(x,t)\rightarrow u(x,t)$
for almost every
$(x,t) \in [0,T]\times \Omega $
. Note that
$f(u_{n_j})(x,t) \rightarrow f(u)(x,t)$
for almost every
$(x,t) \in [0,T]\times \Omega )$
and
$f(u_{n_j})$
is uniformly bounded in
$L^2(0,T;H)$
. Therefore, by Lemma 3.2,
$f(u_{n_j}) \rightharpoonup f(u)$
in
$L^2(0,T;H)$
. This implies that the function
$\chi $
, which is the weak limit of the sequence
$f(u_{n_j})$
, must be equal to
$f(u)$
because there can only be one weak limit in the given function space. Now we have
$u \in L^2(0,T;V)$
and
$ {du}/{dt} \in L^2(0,T;V^*)$
. By Theorem 3.3,
$u \in C^0( [0,T];H)$
.
To show that
$u_n(0)= u(0)$
, let
$\phi \in C^1([0,T];V)$
with
$\phi (T)=0$
. Consider the limiting equation of the approximation
$$ \begin{align*} \bigg\langle \frac{du}{dt} , v \bigg\rangle + a(u,v) = \langle f(u), v \rangle \quad (v \in V). \end{align*} $$
Integrating from
$0$
to T and using integration by parts,
$$ \begin{align} \int^T_0 - \langle u, \phi' \rangle + a(u, \phi) \,dt = \int^T_0 \langle f(u(t)), \phi \rangle \,dt + (u(0),\phi(0)). \end{align} $$
However, from the Galerkin approximation,
$$ \begin{align} \int^T_0 - \langle u_n, \phi' \rangle + a(u_n, \phi) \,dt = \int^T_0 \langle P_n f(u_n(t)), \phi \rangle \,dt + (u_n(0),\phi(0)). \end{align} $$
Recall that
$u_n(0) = P_n u_0 \rightarrow u_0 $
. Taking the limit in (4.7) and comparing it with (4.6) yields
$(u_0 - u(0), \phi (0)) =0 $
, which implies
$u_0 =u(0)$
as
$\phi (0)$
is arbitrary.
To show the uniqueness and continuous dependence of the solutions, take
$u_0, v_0 \in H$
and consider the corresponding solutions
$u,v$
. We define 
. Then, w satisfies
$$ \begin{align*} \frac{dw}{dt} + A w = f(u)-f(v), \quad w(0) = u_0-v_0. \end{align*} $$
We take the inner product with w to obtain
$$ \begin{align*} \frac12 \frac{d\|w\|^2_H}{dt} + (A w,w) = (f(u)-f(v),u-v).\end{align*} $$
Because
$(f(u)-f(v),u-v) \leq c{\|u-v\|_{H}^2}$
and
$ (A w,w) \geq b_{\min } \Vert w \Vert _V $
,
$$ \begin{align*}\frac12 \frac{d\|w\|^2_H}{dt} \leq c\|w\|_H^2.\end{align*} $$
By integrating over t, we get
$\|u(t)-v(t)\|_H \leq \|u_0-v_0\|_H e^{ct}$
, which implies the uniqueness and continuous dependence on initial conditions.
4.2 Strong solutions
In this section, we consider more regular solutions (we refer to such solutions as strong solutions) by regarding (4.1) as an equality that holds in
$L^2(0,T;H)$
.
Theorem 4.2 (Well-posedness of strong solutions).
Equation (4.1) with
$u(0)=u_0 \in V$
has a unique solution u for any
$T>0$
with
$ u \in L^2(0,T;D(A))$
and
${du}/{dt} \in L^2(0,T;H)$
, and
$u \in C^0([0,T];V)$
with the caveat that it may have to be adjusted on a set of measure zero. Furthermore,
$u_0 \mapsto u(t)$
in
$C^0(V;V)$
.
Proof. We follow a similar method as in the proof of Theorem 4.1. We consider the inner product of (4.2) with
$Au_n$
, which gives
$$ \begin{align} \bigg(\frac{d u_n}{dt} , A u_n \bigg) + \| A u_n \|_H^2 = (P_n f(u_n) , A u_n). \end{align} $$
Now, let us assume that
$b_1>b_2$
. By integrating the first term on the left-hand side of (4.8) over
$t\in [0,T]$
, and using the fact that
$({d u_n}/{dt} , A u_n)\ = a(u_n,{d u_n}/{dt} )$
,
$$ \begin{align} \frac{b_{2}}{2}\|u_n(T)\|_{V}^2-\frac{b_{1}}{2}\|u_n(0)\|^2_{V}\leq\int_{0}^{T}\bigg(\frac{du_n}{dt},Au_n\bigg)\,dt. \end{align} $$
However, by applying the Cauchy–Schwarz inequality and Young’s inequality to the right-hand side of (4.8) and combining with (4.9),
$$ \begin{align*} \frac{b_{2}}{2}\|u_n(T)\|_{V}^2-\frac{b_{1}}{2}\|u_n(0)\|^2_{V} +\| u_n \|_{L^2(0,T;D(A))}^2 \leq \frac{1}{2}\|P_nf(u_n)\|_{L^2(0,T;H)}^2+ \frac{1}{2}\| u_n \|_{L^2(0,T;D(A))}^2, \end{align*} $$
which implies
$$ \begin{align*} b_{2}\|u_n(T)\|_{V}^2-b_{1}\|u_n(0)\|^2_{V} \leq \|P_nf(u_n)\|_{L^2(0,T;H)}^2 \leq \|f(u_n)\|_{L^2(0,T;H)}^2. \end{align*} $$
By a similar argument as in the proof of Theorem 4.1, we can show that
$u_n \rightarrow u $
in
$L^2(0,T;D(A))$
and
$P_n f(u_n) \rightharpoonup f(u)$
in
$L^2(0,T;H)$
.
We have
$u \in L^2(0,T; D(A))$
and
${du}/{dt} \in L^2(0,T; H)$
, so if we take
$v = (\nabla u)_i$
, then
$v \in L^2(0,T; V)$
and
${dv}/{dt} \in L^2(0,T; V^*)$
, which allows us to apply Theorem 3.3 to deduce that
$v \in C^0([0,T]; H)$
. Consequently,
$u \in C^0([0,T]; V)$
.
Next, we adapt the continuous uniqueness proof of Theorem 4.1 for V. Take
$u_0, v_0 \in V$
and consider the corresponding solutions
$u,v$
. Define 
. Then, w satisfies
$$ \begin{align*} \frac{dw}{dt} + A w = f(u)-f(v), \quad w(0) = u_0-v_0. \end{align*} $$
By taking the inner product with
$Aw$
and using the fact that f is Lipschitz continuous,
$$ \begin{align*} \bigg(\frac{dw}{dt},Aw\bigg) + \| Aw \|_H^2 \leq (f(u)-f(v),Aw) \leq \frac{1}{2} c^2\|w\|^2_V + \frac{1}{2} \| Aw \|_H^2. \end{align*} $$
We move the second term on the right-hand side to the left-hand side, and drop the
$\| Aw \|_H^2$
term. By integration over
$[0,t]$
and applying similar steps as in (4.9) to the left-hand side,
$$ \begin{align*} \|w(t)\|_{V}^2\leq \frac{c^2}{b_{2}}\int_{0}^{t}\|w(s)\|^2_{V}\,ds+\frac{b_{1}}{b_{2}}\|w(0)\|_{V}^2. \end{align*} $$
From Gronwall’s inequality,
$\|u(t)-v(t)\|_{V}^2\leq (b_{1}/b_{2})\|u(0)-v(0)\|^2_{V}\exp (c^2t/b_{2})$
, which implies uniqueness and continuous dependence on initial conditions. In the case of
$b_2>b_1$
, the result can be obtained by interchanging
$b_2$
and
$b_1$
.
Remark 4.3. Let us consider the classical approach to the Laplacian problem with smooth diffusion and regularise the jump in diffusivity by passing to a limit where the transition region between
$b_1$
and
$b_2$
shrinks to a lower-dimensional surface.
We state a Laplacian problem with diffusion coefficient
$b_{\varepsilon }\colon \bar \Omega \rightarrow \mathbb {R}$
(for all
$\varepsilon>0)$
, where
$b_{\varepsilon }$
is sufficiently smooth. Let
$u_{\varepsilon }$
be the solution of the problem
$$ \begin{align*} \frac{d u_{\varepsilon}}{dt}-\nabla\cdot(b_{\varepsilon}\nabla u_{\varepsilon})&=f \quad\text{in } \Omega\times[0,T], \\ u_{\varepsilon}&=0 \quad\text{on } \partial\Omega\times[0,T], \\ u_{\varepsilon}&=u_{\varepsilon}^{0} \quad\!\!\text{in } \bar{\Omega} \text{ at } t=0, \end{align*} $$
with
$f \in L^2(0,T; L^2(\Omega ))$
. Then,
$u_{\varepsilon }\in L^2(0,T;H^2(\Omega ))\cap L^{\infty }(0,T;H^1_0(\Omega ))$
for all
$\varepsilon>0$
(see [Reference Evans4] for the full statement). We note that the statement can be extended to nonlinear f using the techniques in the proof of Theorem 4.1.
When
$ b_\varepsilon \to b $
as
$ \varepsilon \to 0 $
, the regularised solutions
$ u_\varepsilon $
are uniformly bounded in
$ L^2(0, T; V) $
. By the compactness theorem, we can extract a subsequence that converges weakly in
$ L^2(0, T; V) $
. Hence, by passing to the limit
$ \varepsilon \to 0 $
, the weak limit
$ u $
satisfies the weak formulation of the PDE with the discontinuous diffusion coefficient
$ b $
. Thus, we can obtain
$ u \in L^2(0, T; V) $
. However, the strong solution in
$D(A) $
is not guaranteed due to the potential loss of regularity introduced by the discontinuity in
$ b $
. This is because the higher regularity (that is, second-order derivatives) may not be controlled uniformly as
$ \varepsilon \to 0 $
, particularly at the interface where the jump occurs.
Instead, we tackled the discontinuous diffusion problem directly, proving the existence and uniqueness of weak solutions using the Galerkin method and then leveraging elliptic regularity results to demonstrate that these weak solutions are actually strong solutions.
5 Conclusion
In this paper, we established the global existence and uniqueness of strong solutions for reaction–diffusion equations with diffraction Laplacian and nonlinear terms describing general oxygen consumption. These results extend previous work [Reference de Jong, Prokert and Sterk1] which relied on monotonicity properties of the nonlinear term. This work can be used to make results in [Reference Diez, Krause, Maini, Gaffney and Seirin-Lee3] rigorous as well as provide the theoretical foundation for future numerical work on the dynamics of critical cell states.
Acknowledgements
The authors would like to thank Georg Prokert and Hirofumi Notsu for their assistance.
