Compactness of support of solutions to nonlinear second-order elliptic and parabolic equations in a half-cylinder |
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Authors: | G. V. Grishina |
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Affiliation: | 1. N. é. Bauman Moscow State Technical University, Moscow, Russia
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Abstract: | We study equations of the form $$begin{gathered} u_{tt} + Lu + b(x,t)u_t = a(x,t)left| u right|^{sigma - 1} u, hfill - u_t + Lu = a(x,t)left| u right|^{sigma - 1} u hfill end{gathered}$$ , whereL is a uniformly elliptic operator and 0<σ<1. In the half-cylinder II0,∞={(x, t):x= (x 1,...,x n )∈ ω,t>0}, where ? ? ? n is a bounded domain, we consider solutions satisfying the homogeneous Neumann condition forx∈?ω andt>0. We find conditions under which these solutions have compact support and prove statements of the following type: ifu(x, t)=o(t γ) ast→∞, then there exists aT such thatu(x, t)≡0 fort>T. In this case γ depends on the coefficients of the equation and on the exponent σ. |
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