The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values
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- by Slim Tayachi and Fred B. Weissler PDF
- Trans. Amer. Math. Soc. 366 (2014), 505-530 Request permission
Abstract:
In this paper we prove local existence of solutions of the nonlinear heat equation $u_t = \Delta u + |u|^\alpha u, \; t\in (0,T),\; x\in \mathbb {R}^N,$ with initial value $u(0)=K\partial _{1}\partial _{2}\cdot \cdot \cdot \partial _{m}\delta ,\; K\not =0,\; m\in \{1,\; 2,\; \cdots ,\; N\},\; 0<\alpha <2/(N+m)$ and $\delta$ is the Dirac distribution. In particular, this gives a local existence result with an initial value in a high order negative Sobolev space $H^{s,q}(\mathbb {R}^N)$ with $s\leq -2.$
As an application, we prove the existence of initial values $u_0 = \lambda f$ for which the resulting solution blows up in finite time if $\lambda >0$ is sufficiently small. Here, $f$ satisfies in particular $f\in C_0(\mathbb {R}^N)\cap L^1(\mathbb {R}^N)$ and is anti-symmetric with respect to $x_1,\; x_2,\; \cdots ,\; x_m.$ Moreover, we require $\int _{\mathbb {R}^N} x_1\cdots x_mf(x) dx\not =0$. This extends the known “small lambda” blow up results which require either that $\int _{\mathbb {R}^N}f(x) dx\not =0$ (Dickstein (2006)) or $\int _{\mathbb {R}^N} x_1f(x) dx\not =0$ (Ghoul (2011), (2012)).
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Additional Information
- Slim Tayachi
- Affiliation: Department of Mathematics, Faculty of Science of Tunis, University Tunis El Manar, Campus Universitaire, 2092 Tunis, Tunisia
- MR Author ID: 607511
- Email: slim.tayachi@fst.rnu.tn
- Fred B. Weissler
- Affiliation: Université Paris 13, Sorbonne Paris Cité, CNRS UMR 7539 LAGA, 99, Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France
- Email: weissler@math.univ-paris13.fr
- Received by editor(s): November 21, 2011
- Received by editor(s) in revised form: May 29, 2012
- Published electronically: July 16, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 505-530
- MSC (2010): Primary 35K55, 35A01, 35B44; Secondary 35K57, 35C15
- DOI: https://doi.org/10.1090/S0002-9947-2013-05894-1
- MathSciNet review: 3118404