arXiv subjects

Fred B. Weissler

Publications and source records attributed to Fred B. Weissler.

11 recordsLinked to original sources

arXiv2022

New life-span results for the nonlinear heat equation

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation $\partial_tu-Δu=|u|^αu,\;α>0$ with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of life-span are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for life-span using dilation methods and we give new life-span estimates for Hardy-Hénon parabolic equations.

math.AP↗

arXiv2020

Sign-changing solutions of the nonlinear heat equation with persistent singularities

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = Δu + |u|^αu$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < α<α_0$, where $α_0=\frac {4} {N-4+2\sqrt{ N-1 } }\in (\frac {2} {N-2}, \frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $μ>0$ that can be arbitrarily large, we prove that for any $u_0 \in \mathrm{L} ^\infty _{\mathrm{loc}} ({\mathbb R}^N \setminus \{ 0 \}) $ which is bounded at infinity and equals $μ|x|^{- \frac {2} {α}}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $β|x|^{- \frac {2} {α}}$ at the origin in the sense that for $t>0$, $ |x|^{\frac {2} {α}} u(t,x) \to β$ as $ |x| \to 0$, where $β= \frac {2} {α} ( N -2 - \frac {2} {α} ) $. These solutions in general are neither stationary nor self-similar.

math.AP↗

arXiv2020

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

We consider the nonlinear heat equation $u_t - Δu = |u|^αu$ on ${\mathbb R}^N$, where $α>0$ and $N\ge 1$. We prove that in the range $0 < α<\frac {4} {N-2}$, for every $μ>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= μ|x|^{-\frac {2} {α}}$. 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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arXiv2019

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 - Δu + |u|^αu =0$, where $u=u(t,x)\in {\mathbb R}, $ $(t,x)\in (0,\infty)\times{\mathbb R}^N$ and $α>0$. 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 $u_0 = (-1)^m\partial_1\partial_2 \cdots \partial_m|\cdot|^{-γ} \in {{\mathcal S'}({\mathbb R}^N)}$, $0 < γ< N$. In fact, we show global well-posedness for initial data bounded in an appropriate sense by $u_0$, for any $α>0$. Our approach is to study well-posedness and large time behavior on sectorial domains of the form $Ω_m = \{x \in {{\mathbb R}^N} : x_1, \cdots, x_m > 0\}$, 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 $α$ and $2/(γ+m)$, and we consider all three cases, $α$ equal to, greater than, and less than $2/(γ+m)$. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

math.AP↗

arXiv2018

Perturbations of self-similar solutions

We consider the nonlinear heat equation $u_t = Δu + |u|^αu$ with $α>0$, either on ${\mathbb R}^N $, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) α<4$, for every $μ\in {\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = μ|x-x_0|^{-\frac {2} {α}}$ in a neighborhood of some $x_0\in Ω$ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $μ|x-x_0|^{-\frac {2} {α}}$ on ${\mathbb R}^N $. Moreover, if $μ\ge μ_0$ for a certain $ μ_0( N, α)\ge 0$, and $u_0 I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.

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arXiv2017

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 $u_t = Δu +a |u|^αu, \; t\in(0,T),\; x=(x_1,\,\cdots,\, x_N)\in {\mathbb R}^N,\; a = \pm 1,\; α>0;$ 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 $|u(0)|\leq C(-1)^m\partial_{1}\partial_{2}\cdot \cdot \cdot \partial_{m}(|x|^{-γ})$ for $x_1>0,\; \cdots,\; x_m>0,$ where $C>0$ is a constant, $m\in \{1,\; 2,\; \cdots,\; N\},$ $0<γ 0$ is sufficiently small. We also construct blowing up solutions with initial data $λ_n f$ such that $λ_n^{[({1\over α}-{γ+m\over 2})^{-1}]}T_{\max}(λ_n f)$ has different finite limits along different sequences $λ_n\to 0$. Our result extends the known "small lambda" blow up results for new values of $α$ and a new class of initial data.

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arXiv2015

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

In this paper we consider the nonlinear Schrö\-din\-ger equation $i u_t +Δu +κ|u|^αu=0$. We prove that if $α<\frac {2} {N}$ and $\Im κ<0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $α>\frac {2} {N}$ and $κ\in {\mathbb C}$, 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 }} < α\le \frac {4} {N} $, then small data give rise to global, scattering solutions in $H^1$.

math.AP↗

arXiv2015

Non-regularity in Hölder and Sobolev spaces of solutions to the semilinear heat and Schrödinger equations

In this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $ f ( u ) = λ|u|^αu$. We show that low regularity of $f$ (i.e., $α>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_t= f(w)$. This yields in particular an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for NLS in certain $H^s$ spaces, which depend on the smallness of $α$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel's formula. This yields in particular that if $α$ is sufficiently small and $N$ sufficiently large, then the nonlinear heat equation is ill-posed in $H^s ({\mathbb R}^N ) $ for all $s\ge 0$.

math.AP↗

arXiv2013

Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $ϕ_t = e^{iθ} Δϕ+ e^{iγ} |ϕ|^αϕ$ with periodic boundary conditions. Our result includes all values of $θ$ and $γ$ for which $\cos θ\cos γ>0$, but requires that $α>0$ be sufficiently small.

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arXiv2012

Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-iθ} u_t = Δu+ |u|^α u$ blow up in finite time, where α>0 and π/2<θ<π/2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^θ$ as $θ\to \pm π/2 $. It turns out that $T_{max}^θ$ stays bounded (respectively, goes to infinity) as $θ\to \pm π/2 $ in the case where the solution of the limiting nonlinear Schrödinger equation blows up in finite time (respectively, is global).

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arXiv2012

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.

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