scholarly journals Global Existence and Uniform Energy Decay Rates for the Semilinear Parabolic Equation with a Memory Term and Mixed Boundary Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Zhong Bo Fang ◽  
Liru Qiu
Keyword(s):  
Parabolic Equation ◽  
Mixed Boundary ◽  
Global Solutions ◽  
Energy Functional ◽  
Decay Rates ◽  
Memory Term ◽  
Semilinear Parabolic Equation ◽  
Priori Estimates ◽  
Energy Decay Rates ◽  
Semilinear Parabolic

This work is concerned with a mixed boundary value problem for the semilinear parabolic equation with a memory term and generalized Lewis functions which depends on both spacial variable and time. Under suitable conditions, we prove the existence and uniqueness of global solutions and the energy functional decaying exponentially or polynomially to zero as the time goes to infinity by introducing brief Lyapunov function and precise priori estimates.

Asymptotically self-similar global solutions of a semilinear parabolic equation with a nonlinear gradient term

1999 ◽  
Vol 129 (6) ◽  
pp. 1291-1307 ◽  
Author(s):  
S. Snoussi ◽  
S. Tayachi ◽  
F. B. Weissler
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Asymptotic Behaviour ◽  
Global Solutions ◽  
Gradient Term ◽  
Semilinear Parabolic Equation ◽  
Small Initial Data ◽  
Self Similar ◽  
Semilinear Parabolic

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

scholarly journals Uniform Energy Decay Rates for the Fuzzy Viscoelastic Model with a Nonlinear Source

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fengyun Zhang
Keyword(s):  
Viscoelastic Model ◽  
Energy Decay ◽  
Decay Rates ◽  
Polynomial Form ◽  
Computational Technique ◽  
Exponential Form ◽  
Nonlinear Source ◽  
Priori Estimates ◽  
Energy Decay Rates ◽  
Mathematica Software

This paper considers the fuzzy viscoelastic model with a nonlinear source u t t + L u + ∫ 0 t g t − ζ Δ u ζ d ζ − u γ u − η Δ u t = 0 in a bounded field Ω. Under weak assumptions of the function g t , with the aid of Mathematica software, the computational technique is used to construct the auxiliary functionals and precise priori estimates. As time goes to infinity, we prove that the solution is global and energy decays to zero in two different ways: the exponential form and the polynomial form.

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