Stability of Weak Solutions to Parabolic Problems with Nonstandard Growth and Cross–Diffusion

Erhardt, André H.

doi:10.3390/axioms10010014

Cited by 2 publications

(3 citation statements)

References 21 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

Section: Introductionsupporting

confidence: 66%

“…Similar stability estimates in L 2 (Ω) were proven in [9] for the solutions of parabolic systems with nonstandard growth and a cross-diffusion term. The p(z)-Laplacian is a prototype of the operators with nonstandard growth considered in [7,9]. In [16], the stability estimates in L 1 (Ω) with respect to the initial data were derived for the solutions of anisotropic parabolic equations with double variable nonlinearity, convective terms, and possible degeneracy on the lateral boundary of the problem domain.…”

Section: Introductionsupporting

confidence: 66%

“…9) is fulfilled for q = p * and p = max{p ks , p km }, whence (3.25). (ii) Let us consider problem (3.1) with the exponent p * (•) ∈ C α (Q), p * = lim p km , and the data f , u 0 .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Stability for Evolution Equations with Variable Growth

Shmarev

Simsen

2022

Mediterr. J. Math.

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionsupporting

confidence: 66%