Stability for Evolution Equations with Variable Growth

Abstract

We study the homogeneous Dirichlet problem for the evolution p(x, t)-Laplacian with the nonlinear source $$\begin{aligned} u_t-{\text {div}}\left( |\nabla u|^{p(x,t)-2}\nabla u\right) =f(x,t,u),\quad (x,t)\in Q=\Omega \times (0,T). \end{aligned}$$ u t - div | ∇ u | p ( x , t ) - 2 ∇ u = f ( x , t , u ) , ( x , t ) ∈ Q = Ω × ( 0 , T ) . Here, $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n is a bounded domain, $$n\ge 2$$ n ≥ 2 , and $$p(x,\!t)$$ p ( x , t ) is a given function $$p(\cdot ):Q\mapsto (\frac{2n}{n+2},p^+]$$ p ( · ) : Q ↦ ( 2 n n + 2 , p + ] , $$p^+<\infty $$ p + < ∞ . It is shown that the solution is stable with respect to perturbations of the exponent p(x, t), the nonlinear source f(x, t, u), and the initial datum. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the exponent p(x, t) and the data u(x, 0), f. Estimates on the rate of convergence of solutions of perturbed problems to the solution of the limit problem are derived.

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

“…Dropping the first nonnegative term on the left-hand side of (6.8) and taking into account (3.3) we conclude that 0 ≤ R ǫm (∇u m , ∇u n ) → 0 as m, n → ∞. By [7, Proposition 3.2] with a(z) ≡ 1, b(z) ≡ 0 and ǫ = 0 (see also [31,Lemma 3.8]) we know that (6.10…”

“…To estimate the last term of the second inequality in (6.8) we apply the Lagrange mean value theorem (see the derivation of (3.7)) and then follow the proof of [31,Lemma 3.9]. The constant C depends on ∇u m p(•),QT , ∇u n p(•),QT , which are already estimated in (6.5).…”