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Register a new userSign-changing solutions of the nonlinear heat equation with persistent singularities
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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 $alpha _0=frac {4} {N-4+2sqrt{ N-1 } }in (frac {2} {N-2}, frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $mu >0$ 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 $mu |x|^{- frac {2} {alpha }}$ 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 $beta |x|^{- frac {2} {alpha }}$ at the origin in the sense that for $t>0$, $ |x|^{frac {2} {alpha }} u(t,x) to beta $ as $ |x| to 0$, where $beta = frac {2} {alpha } ( N -2 - frac {2} {alpha } ) $. These solutions in general are neither stationary nor self-similar.
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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 article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter spacetime are created and apparently exist for all times.
In this paper, we study the following Kazdan-Warner equation with sign-changing prescribed function $h$ begin{align*} -Delta u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}}-1right) end{align*} on a closed Riemann surface whose area is equal to one. The solutions are the critical points of the functional $J_{8pi}$ which is defined by begin{align*} J_{8pi}(u)=frac{1}{16pi}int_{Sigma}| abla u|^2+int_{Sigma}u-lnleft|int_{Sigma}he^{u}right|,quad uin H^1left(Sigmaright). end{align*} We prove the existence of minimizer of $J_{8pi}$ by assuming begin{equation*} Delta ln h^++8pi-2K>0 end{equation*}at each maximum point of $2ln h^++A$, where $K$ is the Gaussian curvature, $h^+$ is the positive part of $h$ and $A$ is the regular part of the Green function. This generalizes the existence result of Ding, Jost, Li and Wang [Asian J. Math. 1(1997), 230-248] to the sign-changing prescribed function case. We are also interested in the blow-up behavior of a sequence $u_{varepsilon}$ of critical points of $J_{8pi-varepsilon}$ with $int_{Sigma}he^{u_{varepsilon}}=1, limlimits_{varepsilonsearrow 0}J_{8pi-varepsilon}left(u_{varepsilon}right)<infty$ and obtain the following identity during the blow-up process begin{equation*} -varepsilon=frac{16pi}{(8pi-varepsilon)h(p_varepsilon)}left[Delta ln h(p_varepsilon)+8pi-2K(p_varepsilon)right]lambda_{varepsilon}e^{-lambda_{varepsilon}}+Oleft(e^{-lambda_{varepsilon}}right), end{equation*}where $p_varepsilon$ and $lambda_varepsilon$ are the maximum point and maximum value of $u_varepsilon$, respectively. Moreover, $p_{varepsilon}$ converges to the blow-up point which is a critical point of the function $2ln h^{+}+A$.
We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.
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.
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