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Register a new userThe nonlinear heat equation involving highly singular initial values and new blowup and life span results
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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 $u(0)in L^1_{rm{loc}}left({mathbb R}^Nsetminus{0}right)$, anti-symmetric with respect to $x_1,; x_2,; cdots,; x_m$ and $|u(0)|leq C(-1)^mpartial_{1}partial_{2}cdot cdot cdot partial_{m}(|x|^{-gamma})$ for $x_1>0,; cdots,; x_m>0,$ where $C>0$ is a constant, $min {1,; 2,; cdots,; N},$ $0<gamma<N$ and $0<alpha<2/(gamma+m).$ This gives a local existence result with highly singular initial values. As an application, for $a=1,$ we establish new blowup criteria for $0<alphaleq 2/(gamma+m)$, including the case $m=0.$ Moreover, if $(N-4)alpha<2,$ we prove the existence of initial values $u_0 = lambda f,$ for which the resulting solution blows up in finite time $T_{max}(lambda f),$ if $lambda>0$ is sufficiently small. We also construct blowing up solutions with initial data $lambda_n f$ such that $lambda_n^{[({1over alpha}-{gamma+mover 2})^{-1}]}T_{max}(lambda_n f)$ has different finite limits along different sequences $lambda_nto 0$. Our result extends the known small lambda blow up results for new values of $alpha$ and a new class of initial data.
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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.
We consider the energy critical semilinear heat equation $$ left{begin{aligned} &partial_t u-Delta u =|u|^{frac{4}{n-2}}u &mbox{in } {mathbb R}^ntimes(0,T), &u(x,0)=u_0(x), end{aligned}right. $$ where $ ngeq 3$, $u_0in L^infty({mathbb R}^n)$, and $Tin {mathbb R}^+$ is the first blow up time. We prove that if $ n geq 7$ and $ u_0 geq 0$, then any blowup must be of Type I, i.e., [|u(cdot, t)|_{L^infty({mathbb R}^n)}leq C(T-t)^{-frac{1}{p-1}}.] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysis of bubbling behavior (bubbling tower/cluster).
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 curves are represented by graphs. By using an elementary scaling technique, we show some formulas for space-time behavior of the solution. Keywords: scaling argument, self-similar solution, nonstabilizing solution, nontrivial dynamics, nontrivial large-time behavior, irregular behavior.
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.
We consider the nonlinear Schrodinger equation [ u_t = i Delta u + | u |^alpha u quad mbox{on ${mathbb R}^N $, $alpha>0$,} ] for $H^1$-subcritical or critical nonlinearities: $(N-2) alpha le 4$. Under the additional technical assumptions $alphageq 2$ (and thus $Nleq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^alpha U$. In the simplest case, $U(t,x)=(|x|^k-alpha t)^{-frac 1alpha}$ for $t<0$, $xin {mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|Delta U|ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument.
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