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Critical Exponents for the Heat-Conduction Equation with Nonlocal,Nonlinear Perturbations

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We study the conditions for the existence and nonexistence of global in time (t > 0), nonnegative solutions of the problem

$u_t - \Delta u = u^p \left( {\mathop {\smallint }\limits_{\mathbb{R}^N } u^q (y,t)dy} \right),\quad x \in \mathbb{R}^N ,\quad t > 0,$

$u(x,0) = u_0 (x) \geqslant 0,\quad x \in \mathbb{R}^N ;\quad p > 1,\quad q > 0.$

If p + q ≤ 2 + 2/N, then the problem has no global nontrivial solutions. If p + q > 2 + 2/N, then such solutions exist. Some generalizations of this problem are discussed.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.

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