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In this paper we study global well-posedness and long time asymptotic behavior of solutions to the nonlinear heat equation with absorption, $ u_t - Delta u + |u|^alpha u =0$, where $u=u(t,x)in {mathbb R}, $ $(t,x)in (0,infty)times{mathbb R}^N$ and $alpha>0$. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables $x_1,; x_2,; cdots,; x_m$ for some $min {1,2, cdots, N}$, such as $u_0 = (-1)^mpartial_1partial_2 cdots partial_m|cdot|^{-gamma} in {{mathcal S}({mathbb R}^N)}$, $0 < gamma < N$. In fact, we show global well-posedness for initial data bounded in an appropriate sense by $u_0$, for any $alpha>0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $Omega_m = {x in {{mathbb R}^N} : x_1, cdots, x_m > 0}$, and then to extend the results by reflection to solutions on ${{mathbb R}^N}$ which are antisymmetric. We show that the large time behavior depends on the relationship between $alpha$ and $2/(gamma+m)$, and we consider all three cases, $alpha$ equal to, greater than, and less than $2/(gamma+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.
In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = Delta u +a |u|^alpha u, ; tin(0,T),; x=(x_1,,cdots,, x_N)in {mathbb R}^N,; a = pm 1,; alpha>0;$ with initial value $u(0)in L^1_{rm{loc}}left({mathbb R}^Nsetminu
We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=Delta u^m$ in $({mathbb R}^nsetminus{0})times(0,infty)$ in the subcritical case $0<m<frac{n-2}{n}$, $nge3$. Firstly, we prove the existence of singu
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
We study two initial value problems of the linear diffusion equation and a nonlinear diffusion equation, when Cauchy data are bounded and oscillate mildly. The latter nonlinear heat equation is the equation of the curvature flow, when the moving curv
We study the existence of sign-changing solutions to the nonlinear heat equation $partial _t u = Delta u + |u|^alpha u$ on ${mathbb R}^N $, $Nge 3$, with $frac {2} {N-2} < alpha <alpha _0$, where $alpha _0=frac {4} {N-4+2sqrt{ N-1 } }in (frac {2} {N-