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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values


Authors: Slim Tayachi and Fred B. Weissler
Journal: Trans. Amer. Math. Soc. 366 (2014), 505-530
MSC (2010): Primary 35K55, 35A01, 35B44; Secondary 35K57, 35C15
Published electronically: July 16, 2013
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Abstract: In this paper we prove local existence of solutions of the nonlinear heat equation $ u_t = \Delta u + \vert u\vert^\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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05894-1
PII: S 0002-9947(2013)05894-1
Keywords: Nonlinear heat equation, highly singular initial values, finite time blow--up.
Received by editor(s): November 21, 2011
Received by editor(s) in revised form: May 29, 2012
Published electronically: July 16, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.