Robustness with respect to exponents for nonautonomous reaction–diffusion equations

Samprogna, Rodrigo; Simsen, Jacson

doi:10.14232/ejqtde.2018.1.11

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…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. Continuity of solutions of equation (1.1) with respect to the variable exponent p and convergence to the solution of the limit problem was discussed in papers [12][13][14][15]. In these works, continuity in C([0, T ]; L 2 (Ω)) is proven for the solutions of degenerate equations with p ≡ p(x) > 2.…”

Section: Introductionmentioning

confidence: 99%

Stability for Evolution Equations with Variable Growth

Shmarev

Simsen

2022

Mediterr. J. Math.

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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.

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Section: Introductionmentioning

confidence: 99%

Stability for Evolution Equations with Variable Growth

Shmarev

Simsen

2022

Mediterr. J. Math.

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“…In [16], the stability estimates in 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. Continuity of solutions of equation (1.1) with respect to the variable exponent and convergence to the solution of the limit problem was discussed in papers [15,12,14,13]. In these works, continuity in ([0, ]; 2 (Ω)) is proven for the solutions of degenerate equations with ≡ ( ) > 2.…”

Section: Introductionmentioning

confidence: 99%

Stability for evolution equations with variable growth

Shmarev¹,

Simsen²,

Simsen³

2021

Preprint

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We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution ( , )-Laplacian with the nonlinear sourcewhere Ω is a bounded domain in ℝ , ≥ 2, and ( ,It is shown that the solution is stable with respect to perturbations of the variable exponent ( , ), the nonlinear source term ( , , ), and the initial data. 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 nonlinearity exponent and the data ( , 0), . Estimates on the rate of convergence of a sequence of solutions to the solution of the limit problem are derived.

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Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

Simsen¹

2019

Electron. J. Qual. Theory Differ. Equ.

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Robustness with respect to exponents for nonautonomous reaction–diffusion equations

Cited by 3 publications

References 12 publications

Stability for Evolution Equations with Variable Growth

Stability for Evolution Equations with Variable Growth

Stability for evolution equations with variable growth

Weak upper semicontinuity of pullback attractors for nonautonomous reaction-diffusion equations

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