Asymptotically self-similar global solutions for a higher-order semilinear parabolic equation

2010 ◽  
Vol 33 (12) ◽  
pp. 1473-1481
Author(s):  
Fuqin Sun ◽  
Fan Li ◽  
Xiuqing Jia
Keyword(s):  
Parabolic Equation ◽  
Global Solutions ◽  
Higher Order ◽  
Semilinear Parabolic Equation ◽  
Self Similar ◽  
Semilinear Parabolic

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 Convergence to self-similar solutions for a semilinear parabolic equation

2008 ◽  
Vol 21 (3) ◽  
pp. 703-716 ◽  
Author(s):  
Marek Fila ◽  
◽  
Michael Winkler ◽  
Eiji Yanagida ◽  
◽  
...  
Keyword(s):  
Parabolic Equation ◽  
Semilinear Parabolic Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Semilinear Parabolic

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].

Regional blow-up for a higher-order semilinear parabolic equation

2001 ◽  
Vol 12 (5) ◽  
pp. 601-623 ◽  
Author(s):  
MANUELA CHAVES ◽  
VICTOR A. GALAKTIONOV
Keyword(s):  
Parabolic Equation ◽  
Evolution Equations ◽  
Blow Up ◽  
Semilinear Parabolic Equation ◽  
General Integral ◽  
Superlinear Growth ◽  
Self Similar Solution ◽  
Self Similar ◽  
Semilinear Parabolic ◽  
Hamilton Jacobi Equation

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equation[formula here]with a superlinear function q(u) for |u| Gt; 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u|ln|u‖2m. We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T− is described by the self-similar solution[formula here]of the complex Hamilton–Jacobi equation[formula here].

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.

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