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Variational problems related to self-similar solutions of the heat equation. (English) Zbl 0639.35038
We study variational problems related to the existence of self similar solutions of \(u_ t-\Delta u+\epsilon | u|^{p-1}u=0\) where \(\epsilon =\pm 1\), \(p>1\), u(t,x)\(\in R\) m, \(m\geq 1\). To this end we prove some compactness results for the embedding \(H\quad 1(K_{\theta})\subset L^ 2(K_{\theta})\) where H \(1(K_{\theta})\) is a weighted Sobolev space, and we apply classical variational methods to prove existence or non-existence of solutions for equations such as -\(\nabla f-\nabla \theta \cdot \nabla f+\epsilon | f|^{p-1}f=\lambda f\), depending on the values of \(\epsilon\),p and \(\lambda\in R\).
Reviewer: M.Escobedo

MSC:
35K55 Nonlinear parabolic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A15 Variational methods applied to PDEs
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