scholarly journals Asymptotic Behavior and Convergence of Solutions of a Semilinear Transport Equation with Delay

2001 ◽  
Vol 254 (2) ◽  
pp. 464-483 ◽  
Author(s):  
He Mengxing ◽  
Luo Ronggui
Keyword(s):  
Asymptotic Behavior ◽  
Transport Equation ◽  
Convergence Of Solutions ◽  
Equation With Delay

Well-Posedness and Asymptotic Behavior of a Nonclassical Nonautonomous Diffusion Equation with Delay

2015 ◽  
Vol 25 (14) ◽  
pp. 1540021 ◽  
Author(s):  
Tomás Caraballo ◽  
Antonio M. Márquez-Durán ◽  
Felipe Rivero
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Diffusion Equation ◽  
Existence Of Solutions ◽  
Local Solution ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Pullback Attractors ◽  
Nonautonomous Dynamical Systems ◽  
Equation With Delay

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzelà theorem to prove the compactness.

scholarly journals Blow up and asymptotic behavior of solutions for a p(x)-Laplacian equations with delay term and variadle exponents

2021 ◽  
Author(s):  
Stanilslav Antontsev ◽  
Jorge Ferreira ◽  
Erhan Pişkin ◽  
Hazal Yüksekkaya
Keyword(s):  
Asymptotic Behavior ◽  
Integral Inequality ◽  
Blow Up ◽  
Asymptotic Behavior Of Solutions ◽  
Variable Exponents ◽  
Delay Term ◽  
Laplacian Equation ◽  
Laplacian Equations ◽  
Equation With Delay ◽  
Equations With Delay

In this paper, we consider a nonlinear p .x/Laplacian equation with delay of time and variable exponents. Firstly, we prove the blow up of solutions. Then, by applying an integral inequality due to Komornik, we obtain the decay result. These results improve and extend earlier results in the literature.

scholarly journals Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-23
Author(s):  
Aboubacar Marcos ◽  
Ambroise Soglo
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Variable Exponent ◽  
Probability Space ◽  
Compact Embedding ◽  
Convergence Of Solutions ◽  
Existence Of Positive Solutions ◽  
Wasserstein Space ◽  
Laplacian Equations

In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation ∂ρt,x/∂t=divxρt,x∇xG′ρ+Vqx−2∇xG′ρ+V. The potential V is not necessarily smooth but belongs to a Sobolev space W1,∞Ω. Given the initial datum ρ0 as a probability density on Ω, we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding W1,q.Ω↪↪Lq.Ω established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space.

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