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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.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
45K05 Integro-partial differential equations
92D30 Epidemiology