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Exponential decay and stability of Volterra diffusion equations. (English) Zbl 0958.45004
Corduneanu, C. (ed.) et al., Volterra equations and applications. Proceedings of the Volterra centennial symposium, University of Texas, Arlington, TX, USA, May 23-25, 1996. London: Gordon and Breach Science Publishers. Stab. Control Theory Methods Appl. 10, 299-307 (2000).
From the introduction: We consider the following parabolic equation of finding $$u=u(x,t)$$ such that $u_t+Au+ \int^t_0K(t-s)Bu(s)ds +f(u)=0, \text{ in }Q_T, \quad u=0,\text{ on }\partial \Omega\times (0,T],\tag{1}$ with the nonlocal time weighting initial condition $u(x,0)= \sum^M_{k=1} \beta_k(x)u(x,T_k) +\psi(x),\;x\in\Omega. \tag{2}$ where $$Q_T=\Omega\times (0,T]$$, $$\Omega\subset \mathbb{R}^d$$ $$(d\geq 1)$$ is an open bounded domain with smooth boundary $$\partial\Omega$$, $$T>0$$ and $$0<T_1<T_2 <\cdots <T_M=T$$, $$\beta_k(x)$$, $$\psi(x)$$ and $$f(u)$$ are known smooth functions with respect to their variables, and $$A$$ is a strongly elliptic operator $A=-\sum^d_{i,j=1} {\partial\over \partial x_j}\left(a_{i,j} (x){\partial \over\partial x_i}\right) +a(x),\;x\in \Omega, \tag{3}$ and $$B$$ is any second order operator $B=-\sum^d_{i,j=1} {\partial\over \partial x_j}\left(b_{i,j}(x) {\partial\over\partial x_i}\right)+ \sum^d_{i=1} {\partial\over \partial x_i}\left(b_i(x) {\partial\over \partial x_i}\right) +b(x),\;x\in\Omega,$ where $$a(x)$$, $$a_{i,j}(x)= a_{j,i}(x)$$, $$b(x)$$, $$b_i(x)$$, $$b_{i,j}(x)$$ are known smooth functions, and satisfy for some positive constants $$a_0$$, $$a_1>0$$ $a_0\xi^2\leq \sum^d_{i,j=1} a_{i,j}\xi_i \xi_j\leq a_1\xi^2, \quad x\in\Omega,\;\xi\in \mathbb{R}^d.$ In this note we report the results obtained by Y. Lin and J. Lin [Stability of diffusion equations with integrals and non-local initial conditions, preprint, Department of Mathematics, University of Alberta (1996)] for the problem (1)–(2) regarding the well-posedness, exponential decay as $$t\to\infty$$ (if $$\beta_k=0)$$ and its time-discretization by backward Euler difference methods.
When $$\beta_k\equiv 0$$ and $$f=0$$ for $$k=1,2,\dots M$$, backward Euler and second order Euler schemes in finite element settings were analyzed by V. Thomee and L. B. Wahlbin [Math. Comput. 62, No. 206, 477-496 (1994; Zbl 0801.65135)], where certain exponential decay properties were obtained under some strong assumptions on the kernel. In this paper we study the exponential decay not only for the real continuous solutions but also for its numerical solutions. The exponential decay rate obtained is optimal. That is, it reduces to classical results for parabolic equations if the kernel $$K(t) =0$$. In addition finite difference schemes of backward Euler methods also are considered and shown to decay exponentially.
For the entire collection see [Zbl 0940.00038].
##### MSC:
 45J05 Integro-ordinary differential equations 45M10 Stability theory for integral equations 65R20 Numerical methods for integral equations