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On some mixed boundary value problems with nonlocal diffusion. (English) Zbl 1064.35083
From the introduction: Let $$\Omega$$ be a bounded open set of $$\mathbb{R}^n$$, $$n\geq 1$$ with a Lipschitz boundary $$\Gamma$$. We suppose that $$\Gamma$$ is split into two measurable subsets $$\Gamma_D$$ and $$\Gamma_N=\Gamma\setminus\Gamma_D$$. We denote by $$a=a(\zeta)$$ a function such that $\begin{cases} a\text{ is continuous},\\ \exists m,M\text{ such that }0<m\leq a(\zeta)\leq M\quad\forall\zeta\in\mathbb{R}.\end{cases}$ We consider then the problem of finding $$u=u(x,t)$$ solution to $\begin{cases} u_t-a \biggl(\ell\bigl(u(t)\bigr)\biggr)\Delta u+u=f\quad & \text{in }\Omega \times \mathbb{R}^+,\\ u=0\text{ on }\Gamma_D\times\mathbb{R}^+,\frac{\partial u}{\partial\nu}=0 \quad &\text{on }\Gamma_N\times\mathbb{R}^+,\\ u( \cdot,0)=u_0\text{ in }\Omega. \end{cases}$ In the above system $$\ell$$ is a linear form on $$L^2(\Omega)$$ so that $$\ell(u(t))=\int_\Omega g(x) u(x,t)\,dx$$, $$g\in L^2(\Omega)$$, $$u_0$$ and $$f$$ are some functions such that $$f\in L^2(\Omega)$$, $$u_0\in L^2(\Omega)$$. This kind of model problem arises for instance in diffusion of bacteria: $$u(x,t)$$ is the density of population located at $$x$$ at the time $$t$$, $$f$$ is the density of bacteria supplied from outside, $$u_0$$ is the initial density of population, $$a$$ is the diffusion rate (depending on $$\ell(u(t)))$$, the lower order term $$u$$ is the density of population eliminated by death at a constant rate taken for the sake of simplicity equal to 1.

MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 45K05 Integro-partial differential equations 92D30 Epidemiology
Keywords:
diffusion of bacteria