The nonlinear heat equation involving highly singular initial values and new blowup and life span results

In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = \Delta u +a |u|^\alpha u, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; \alpha>0;$ with initial value $u(0)\in L^1_{\rm{loc}}\left({\mathbb R}^N\setminus\{0\}\right)$, anti-symmetric with respect to $x_1,\; x_2,\; \cdots,\; x_m$ and $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-\gamma})$ for $x_1>0,\; \cdots,\; x_m>0,$ where $C>0$ is a constant, $m\in \{1,\; 2,\; \cdots,\; N\},$ $0<\gamma<N$ 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 $0<\alpha\leq 2/(\gamma+m)$, 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 $\lambda_n^{[({1\over \alpha}-{\gamma+m\over 2})^{-1}]}T_{\max}(\lambda_n f)$ has different finite limits along different sequences $\lambda_n\to 0$. Our result extends the known "small lambda" blow up results for new values of $\alpha$ and a new class of initial data.Comment: Submitte

Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi$ with periodic boundary conditions. Our result includes all values of $\theta$ and $\gamma$ for which $\cos \theta \cos \gamma >0$, but requires that $\alpha >0$ be sufficiently small

Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where $\alpha >0$ and $N\ge 1$. We prove that in the range $0 0$, 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

A Fujita-type blowup result and low energy scattering for a nonlinear Schr\"o\-din\-ger equation

In this paper we consider the nonlinear Schr\"o\-din\-ger equation $i u_t +\Delta u +\kappa |u|^\alpha u=0$. We prove that if $\alpha <\frac {2} {N}$ and $\Im \kappa <0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $\alpha >\frac {2} {N}$ and $\kappa \in {\mathbb C}$, we improve the existing low energy scattering results in dimensions $N\ge 7$. More precisely, we prove that if $\frac {8} {N + \sqrt{ N^2 +16N }} < \alpha \le \frac {4} {N}$, then small data give rise to global, scattering solutions in $H^1$

Self-Similar Subsolutions and Blowup for Nonlinear Parabolic Equations

AbstractFor a wide class of nonlinear parabolic equations of the formut−Δu=F(u,∇u), we prove the nonexistence of global solutions for large initial data. We also estimate the maximal existence time. To do so we use a method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results onut−Δu=up, on generalized Burgers' equations, and on other semilinear equations. This method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems, both semilinear and quasilinear

Large time behavior of solutions to the nonlinear heat equation with absorption with highly singular antisymmetric initial values

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 $m\in \{1,2, \cdots, N\}$, such as $u_0 = (-1)^m\partial_1\partial_2 \cdots \partial_m|\cdot|^{-\gamma} \in {{\mathcal S'}({\mathbb R}^N)}$, $0 < \gamma < N$. 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

Singularity formation and blowup of complex-valued solutions of the modified KdV equation

The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$u_t + 6u^2u_x + u_{xxx} = 0$$ are determined. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time

Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^\theta$ as $\theta \to \pm \pi /2$. It turns out that $T_{max}^\theta$ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi /2$ in the case where the solution of the limiting nonlinear Schr\"odinger equation blows up in finite time (respectively, is global).Comment: 22 page