2025/07/07 by Arora, Rakesh, Shmarev, Sergey
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.04924
We investigate the homogeneous Dirichlet problem for the irregular double-phase evolution equation ut-div ( a(z)|∇ u|p(z)-2 ∇ u + b(z)|∇ u|q(z)-2 ∇ u)=f(z), z=(x,t)∈ QT:=Ω× (0,T), where Ω⊂ ℝN, N ≥ 2 is a bounded domain, T>0, The non-differentiable coefficients a(z), b(z), the free term f, and the variable exponents p, q are given functions. The coefficients a and b are nonnegative, bounded, satisfy the inequality a(z)+b(z)≥ α in QT, and |∇ a|, |∇ b|, at, bt ∈ Ld(QT) for some constant α>0, and with d>2 depending on sup p(z), sup q(z), N, and the regularity of initial data u(x,0). The free term f and initial data u(x,0) satisfy f∈ Lσ(QT) with σgt;2 and |∇ u(x,0)|∈ Lr(Ω) with r≥ max \2,supQTp(z),supQTq(z)\. The variable exponents p,q ∈ C0,1(QT) satisfy the balance condition (2N)/(N+2) lt; p(z), q(z)lt; +∞ in QT and max QT|p(z)-q(z)|lt; \dfrac2N+2. Under the above assumptions, we establish the existence of a solution, which is obtained as the limit of classical solutions to a family of regularized problems and preserves initial temporal integrability: |∇ u(⋅, t)| ∈ Lr(Ω) for a.e. t ∈ (0,T), gains global higher integrability: |∇ u|^min\p(z), q(z)\ + s +r ∈ L1(QT) for any s ∈ (0, (4)/(N+2)), and attains second-order regularity: a(z) |∇ u|(p+r-2)/(2)+b(z) |∇ u|(q+r-2)/(2)∈ L2(0,T;W1,2(Ω)).