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Register a new userOn parabolic convergence of positive solutions of the heat equation
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Authors
Jayanta Sarkar
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In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.
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In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the parabolic limit of a positive solution $u$ at a point on the boundary is the existence of the strong derivative of the boundary measure of $u$ at that point. Moreover, the parabolic limit and the strong derivative are equal.
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We consider the nonlinear heat equation $u_t - Delta u = |u|^alpha u$ on ${mathbb R}^N$, where $alpha >0$ and $Nge 1$. We prove that in the range $0 < alpha <frac {4} {N-2}$, for every $mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= mu |x|^{-frac {2} {alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.
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