Existence of weak solutions of parabolic systems with p, q-growth

Abstract

We consider evolutionary problems associated with a convex integrand f : T × R N n → [0, ∞), which is α-Hölder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions u : T → R N , which solve ∂ t u − div ∂ ζ f (x, t, Du) = 0 weakly in T. Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q − p, namely 2n n + 2 < p ≤ q < p + 1. For 2n n + 2 < p ≤ q < p + min{2, p} α n + 2 , we prove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.

On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p,q)-growth

“…In the parabolic case, in contrast to the elliptic setting, not much is known for the non-standard (p, q)-growth condition. We refer to [4,5,34,35] and references therein for some results about regularity in parabolic problems with (p, q)-growth. As far as we know, the case of discontinuous coefficients has not been exploited yet.…”

“…with A h and B h as introduced in (3.8). From (3.23) and (1.7), we have 34) where D(h) and Γ(h) are defined in (3.12) and (3.14), respectively. Next, we integrate the inequality (3.33) over the parabolic cylinder Q R/4 and use Hölder's inequality with exponents 2(q − 1)/p and 2(q − 1)/(p − 2 + 2(q − p)).…”

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

“…[14,17,18,22,23] for an overview of the state of the art on this matter and [2,3], where more general structures are analyzed. Finally, the question of existence of regular solutions of (1.1) when the nonlinear tensor a(•) has unbalanced polynomial growth was treated in [8,9,31,32]. The theory exposed in these papers confirms that, as in the elliptic case, a restriction like (1.6) on the ratio q/p suffices to prove existence of regular solutions to (1.1).…”

“…Our main goal it to prove that the sequence {v j } is bounded, uniformly with respect to j ∈ N in the space-time L p -norm. Since this is quite a routine procedure, we will just sketch it and refer the reader to [8,31], for more details. Modulo using Steklov averages, we can test (3.13) against the difference v j − f j to get…”

Equivalence between viscosity and weak solutions for the parabolic equations with nonstandard growth

“…Very recently, De Filippis 6 established the gradient bounds for solutions to irregular parabolic equations with ( p , q )‐growth. We refer to Marcellini and Singer 7‐9 and references therein for more results.…”