Gradient bounds for solutions to irregular parabolic equations with $(p,q)$-growth

Filippis, Cristiana De

doi:10.48550/arxiv.2004.01452

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2020

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“…Equation (1.1) with constant exponents p and q furnishes a prototype of the equations recently studied in papers [9,13,16,23] in the context of weak or variational solutions. The proofs of existence also use the local higher integrability of the gradient, but for the existence of variational solutions a weaker assumption on the gap q − p is required.…”

Section: Previous Workmentioning

confidence: 99%

Strong solutions of the double phase parabolic equations with variable growth

Arora,

Shmarev

2020

Preprint

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This paper addresses the questions of existence and uniqueness of strong solutions to the homogeneous Dirichlet problem for the double phase equation with operators of variable growth:wheregiven nonnegative coefficient, and the nonlinear source term has the formThe variable exponents p, q, σ are given functions defined on Q T , p, q are Lipschitz-continuous andWe find conditions on the functions f 0 , a, b, σ and u 0 sufficient for the existence of a unique strong solution with the following global regularity and integrability properties:The same results are established for the equation with the regularized flux∇u + a(z)(ǫ 2 + |∇u| 2 ) q(z)−2 2 ∇u, ǫ > 0.

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Section: Previous Workmentioning

confidence: 99%