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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)
45K05 Integro-partial differential equations
45M10 Stability theory for integral equations
65R20 Numerical methods for integral equations