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Irregular Double-Phase Evolution Problem: Existence and Global Regularity
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DOI：10.1007/s12220-026-02421-0.png)
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Abstract
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We study the homogeneous Dirichlet problem for the double-phase evolution equation u(t) - div(a(z)|del u|(p(z)-2)del u+b(z)|del u|(q(z)-2)del u)del u = f(z) z = (x, t) is an element of Q(T) = Omega & times; (0,T). The non-differentiable coefficients a(z), b(z) and the variable exponents p(z), q(z) are given functions. The coefficients a, b are nonnegative and bounded, with |del a|, |del b|, a(t), b(t) is an element of L-d(Q(T)), d > 2, and such that a(z) + b(z) >= alpha > 0. It is shown that if u(0) is an element of W-0(1,r) (Omega) with a sufficiently large r and f is an element of LN+2(Q(T)), then u(., t) is an element of W-0(1,r) (Omega) for a.e. t is an element of (0, T), |del u|(min{p(z),q(z)}+s+r) is an element of L-1(Q(T)) for any s is an element of (0, 4/N+2), and a(z)|del u|(p(z)+r-2/2) +b(z)|del u|(q(z)+r-2/2) is an element of L-2(0, T; W-1,W-2(Omega)). The case f is an element of L-sigma(Q(T)) with sigma is an element of (2, N+2)} is also studied.
Keywords:
Nonlinear parabolic equation
Double-phase
Global second-order regularity
Gradient integrability
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1.5
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304
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