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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 \(\gamma=\gamma(A,a,N)>0\).

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K05 Heat equation
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