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Heat equation on noncompact Riemannian manifolds. (Russian) Zbl 0743.58031
Let $$(M^ n,g)$$ be open, complete, $$\Delta$$ the Laplace operator acting on functions, $$u_ t-\Delta u=0$$ the heat equation. The author investigates the Harnack inequality for positive solutions of the heat equation consisting in an estimate for $$\sup_{B_ r} u/\inf_{B_ r} u$$. Consider the conditions a) and b). a) $$\hbox{vol}(B_{2R})/\hbox{vol}(B_ R)\leq A$$, $$A$$ independent of the center. b) For any function $$f\in C^ \infty(B^ x_{NR})$$ $\int_{B^ x_{NR}}|\nabla f|^ 2\geq (a/R^ 2)\inf_{\xi\in\mathbb{R}}\int_{B^ x_{NR}}(f-\xi)^ 2,$ where $$N$$ is a certain number $$>1$$. Let $$z\in M$$, $$B_ R=B^ z_ R$$, $$Z_ R=B_ R\times]0,R^ 2[$$. Then the author proves the following theorem. Assume that $$M$$ satisfies the conditions a) and b). Let $$u(x,t)$$ be a positive solution of the heat equation in $$Z_{8r}$$, smooth in $$\bar Z_{8R}$$ and satisfying a Neumann condition at $$x\in\partial M$$ if $$\partial M\neq \emptyset$$. Set $$\tilde Z=B_ R\times ]3R^ 2,4R^ 2[$$ and assume $$\sup_ Z u=1$$. Then $$u(z,64R^ 2)\geq \gamma$$, where $$\gamma=\gamma(A,a,N)>0$$.

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35K05 Heat equation
##### Keywords:
heat equation; Harnack inequality
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