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Register a new userBoundary behavior of positive solutions of the heat equation on a stratified Lie group
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Authors
Jayanta Sarkar
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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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The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky, Crelle, 80:1-32, 1875) that a solution to the heat equation may not be time-analytic at $t=0$ even if the initial function is real analytic. Recently, it was shown in cite{Zha20, DZ20, DP20} that solutions to the heat equation in the whole space, or half space with zero boundary value, are analytic in time under essentially optimal conditions. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any $delta>0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+delta$, which is nowhere analytic in time.
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.
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.
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary condition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$Omega = (-1,1)timesmathbb{T}timesmathbb{T}$$ taking as observation regions slices of the form $omega=(a,b) times mathbb{T} times mathbb{T}$ or tubes $omega = (a,b) times omega_y times mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $omega$ and the boundary of $Omega$: $$T_{min} geqslant frac{1}{8} min{(1+a)^2,(1-b)^2}.$$ Our proof follows a mixed strategy which combines the approach by Lebeau and Robbiano, which relies on Fourier decomposition, with Carleman inequalities for the heat equations that are solved by the Fourier modes. We extend the analysis to the unbounded domain $(-1,1)timesmathbb{T}timesmathbb{R}$.
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