We study the homogeneous Dirichlet problem for the double-phase evolution equation
z = (x, t) ∈ Q_T = Ω × (0, T ), Ω ⊂ R^N , N ≥ 2.
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 and such that in Q_{T, .}
It is shown that if u₀ ∈ W₀^1,r (Ω) with r ≥ max{2, max p(z), max q(z)}, f ∈ L^{N +2}(Q_T ), max |p(z) − q(z)| < , then the problem admits a unique solution with the following properties:
• The solution preserves the initial regularity, for ;
• The gradient acquires higher integrability, |∇u|^{min{p(z),q(z)}+s+r} ∈ L1(Q_T ) for any ;• The solution possesses the second-order regularity,.
The regularity properties remain true in the case f ∈ L^σ (Q_T ) with σ ∈ (2, N + 2), but the admissible values of r are in a bounded interval depending on N and σ.
This is a joint project [1, 2] with Dr. Rakesh Arora from the Indian Institute of Technology at Varanasi, India.
[1] R.Arora, S.Shmarev “Global gradient estimates for solutions of parabolic equations with nonstandard growth”. J. Math. Anal. Appl. 549 (2), 129582, 36 pp., (2025). DOI 10.1016/j.jmaa.2025.129582
[2] R.Arora, S.Shmarev “Irregular double-phase evolution problem: Existence and global regularity” ArXiv, 41 pp., 2025, https://arxiv.org/abs/2507.04924