Existence of Weak Solutions of the Aggregation Equation with the p(·)-Laplacian

Abstract

We consider an elliptic-parabolic aggregation equation of the form

$$ b{(u)}_t=\operatorname{div}\left({\left|\nabla u\right|}^{p(x)-2}\nabla u-b(u)G(u)\right)+\gamma \left(x,b(u)\right), $$

where b is a nondecreasing function and G(u) is an integral operator. The condition on the boundary of a bounded domain Ω ensures that the mass of the population ∫u(x, t)dx = const is preserved for γ = 0. We prove the existence of a weak solution of the problem with a nonnegative bounded initial function in the cylinder Ω × (0, T). A formula for the guaranteed time T of the existence of the solution is obtained.

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Correspondence to V. F. Vildanova.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.

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Vildanova, V.F., Mukminov, F.K. Existence of Weak Solutions of the Aggregation Equation with the p(·)-Laplacian. J Math Sci 252, 156–167 (2021). https://doi.org/10.1007/s10958-020-05150-z

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Keywords and phrases

  • aggregation equation
  • p(·)-Laplacian
  • existence of solution

AMS Subject Classification

  • 35K20
  • 35K55
  • 35K65