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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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