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On the asymptotic behavior of solutions of parabolic equations. (English) Zbl 0534.35051
We consider the asymptotic behavior of solutions u(x,t) of \(-u_ t+L_ xu=-f(x,t)\) in the cylinder \(\Omega\times(0,\infty)\) where \(L_ x\) is an elliptic operator, and show that under the assumption \(\int_{\Omega}| f(x,t)|^ p\quad dx\to 0\) as \(t\to \infty\) for p sufficiently large, the solutions satisfying homogeneous Dirichlet or Robin boundary conditions tend uniformly to zero as \(t\to \infty\); while the solutions satisfying a homogeneous Neumann boundary condition tend uniformly to constants a \(t\to \infty\) provided that \(\lim_{t\to \infty}\int^{t}_{0}\int_{\Omega}f(x,\tau)dx d\tau\) exists.

MSC:
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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