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Stability of nonlocal diffusion equations. (English) Zbl 0945.45004
The partial integro-differential equation $u_t + Au + \int_0^t K(t-s) Bu(s)ds + f(u) = 0, \quad 0 < t \leq T,$ is studied on a smooth domain $$\Omega$$ with zero boundary conditions. The operator $$A$$ is a strongly elliptic operator, so that the equation is parabolic in the absence of the integral term. The operator $$B$$ is a second order differential operator with respect to $$x$$, and $$K$$ is a scalar nonnegative kernel. A number of theorems is proved on exponential decay and time-discretization by backward Euler difference methods. In addition, the well-posedness and time-discretization of this problem are studied under a nonstandard non-local time weighted initial condition $u(x,0) = \sum_{k=1}^M \beta_k(x)u(x,T_k) + \psi(x), \quad x \in \Omega,$ where $$0 < T_1 < T_2 \cdots < T_M = T$$.
Reviewer: O.Staffans (Åbo)
##### MSC:
 45K05 Integro-partial differential equations 45M10 Stability theory for integral equations 65R20 Numerical methods for integral equations