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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.
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
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
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 s
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 pro
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 $a