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Global and exploding solutions for nonlocal quadratic evolution problems. (English) Zbl 0940.35035

The authors consider the local and global solvability of integro-differential parabolic equations arising in statistical mechanics. The diffusive term can be represented by a fractional power of the Laplacian. The conditions for global in time existence and finite time blow-ups are studied. For certain homogeneous initial data the self-similar solutions are constructed. The authors pay special attention to some important cases: Brownian diffusion (local existence, uniqueness and positivity), fractal diffusion case (local existence). If some additional conditions on the decay of the integral multiplayer of the potential type are imposed, then the local solution is extended to the global one. It is shown that for a potential kernel, corresponding to the gravitational interaction of particles, the solution cannot be global in time provided that the inital condition has a sufficiently big integral. Also, existence of mild self-similar solutions along with their asymptotic behavior, and some other results that specify the mentioned theorems are presented.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
82C21 Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q72 Other PDE from mechanics (MSC2000)
45K05 Integro-partial differential equations
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