scholarly journals Error Estimates for Solutions of the Semilinear Parabolic Equation in Whole Space

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaomei Hu
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Error Estimates ◽  
Three Dimensional ◽  
Semilinear Parabolic Equation ◽  
Energy Methods ◽  
Analysis Technique ◽  
Whole Space ◽  
Fourier Analysis Technique ◽  
Semilinear Parabolic

This paper is focused on the error estimates for solutions of the three-dimensional semilinear parabolic equation with initial datau0∈L2(ℝ3). Employing the energy methods and Fourier analysis technique, it is proved that the error between the solution of the semilinear parabolic equation and that of linear heat equation has the behavior asO((1+t)−3/8).

scholarly journals Construction of type I blowup solutions for a higher order semilinear parabolic equation

2019 ◽  
Vol 9 (1) ◽  
pp. 388-412 ◽  
Author(s):  
Tej-Eddine Ghoul ◽  
Van Tien Nguyen ◽  
Hatem Zaag
Keyword(s):  
Parabolic Equation ◽  
Index Theory ◽  
Classical Case ◽  
Higher Order ◽  
Semilinear Parabolic Equation ◽  
Type I ◽  
Finite Dimensional ◽  
Whole Space ◽  
Linearized Operator ◽  
Semilinear Parabolic

Abstract We consider the higher-order semilinear parabolic equation $$\begin{array}{} \displaystyle \partial_t u = -(-{\it\Delta})^{m} u + u|u|^{p-1}, \end{array}$$ in the whole space ℝN, where p > 1 and m ≥ 1 is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m = 1, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [15].

scholarly journals AlgebraicL2Decay for Weak Solutions of the Nonlinear Heat Equations in Whole SpaceR3

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Yuexing Yang
Keyword(s):  
Weak Solutions ◽  
Nonlinear Term ◽  
Heat Equations ◽  
Energy Methods ◽  
Time Decay ◽  
Analysis Technique ◽  
Nonlinear Heat Equations ◽  
Whole Space ◽  
Fourier Analysis Technique ◽  
Time Decay Rate

We obtained the algebraicL2time decay rate for weak solutions of the nonlinear heat equations with the nonlinear term∇u2uin whole spaceR3. The methods are based on energy methods and Fourier analysis technique.

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.

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