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On a nonlinear integro-differential equation of parabolic type. (Russian) Zbl 0575.45012

Let \(\Omega\) be a bounded domain of \(R^ n\), and let \(Q=\Omega \times (0,T)\), \(0<T<\infty\). The author is concerned with the existence of generalized solutions of the initial-boundary value problem \(\partial u/\partial t-\sum^{n}_{i=1}(\partial /\partial x_ i)[a(s)\partial u/\partial x_ i]=f(x,t)\), \((x,t)=(x_ 1,...,x_ n,t)\in Q\), \(u(x,t)=0\), (x,t)\(\in \partial \Omega \times (0,T)\), \(u(x,0)=0\), \(x\in \Omega\). Here \(s(x,t)=\int^{t}_{0}(\text{grad} u(x,r))^ 2dr\), grad u\(=(\partial u/\partial x_ 1,...,\partial u/\partial x_ n)\), \(a(s)=(1+s)^ p\), \(0<p\leq 1\), and f is a given real function on Q, satisfying f, \(\partial f/\partial t\in L^ 2(Q)\), and \(f(x,0)=0\). The approach is based on the use of Galerkin approximations.
Reviewer: S.Aizicovici

MSC:

45K05 Integro-partial differential equations
45L05 Theoretical approximation of solutions to integral equations
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