Global solutions to evolution equations of parabolic type

1986 ◽  
pp. 254-266 ◽  
Author(s):  
Wolf von Wahl
Keyword(s):  
Evolution Equations ◽  
Parabolic Type ◽  
Global Solutions

scholarly journals Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun
Keyword(s):  
Evolution Equations ◽  
Global Solutions ◽  
Global Attractors ◽  
Nonlinear Evolution Equations ◽  
Time Behavior ◽  
Asymptotic Regularity ◽  
Compact Attractor ◽  
Long Time ◽  
Critical Exponential Growth ◽  
Semilinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

scholarly journals Flow invariance for perturbed nonlinear evolution equations

1996 ◽  
Vol 1 (4) ◽  
pp. 417-433 ◽  
Author(s):  
Dieter Bothe
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Accretive Operator ◽  
Nonlinear Evolution Equations ◽  
Evolution System ◽  
Reaction Diffusion Systems ◽  
Closed Sets

LetXbe a real Banach space,J=[0,a]⊂R,A:D(A)⊂X→2X\ϕanm-accretive operator andf:J×X→Xcontinuous. In this paper we obtain necessary and sufficient conditions for weak positive invariance (also called viability) of closed setsK⊂Xfor the evolution systemu′+Au∍f(t,u)  on  J=[0,a]. More generally, we provide conditions under which this evolution system has mild solutions satisfying time-dependent constraintsu(t)∈K(t)onJ. This result is then applied to obtain global solutions of reaction-diffusion systems with nonlinear diffusion, e.g. of typeut=ΔΦ(u)+g(u)  in  (0,∞)×Ω,   Φ(u(t,⋅))|∂Ω=0,   u(0,⋅)=u0under certain assumptions on the setΩ⊂Rnthe functionΦ(u1,…,um)=(φ1(u1),…,φm(um))andg:R+m→Rm.

scholarly journals Nonexistence of global solutions for a class of viscoelastic wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge A. Esquivel-Avila
Keyword(s):  
Wave Equations ◽  
Evolution Equations ◽  
Blow Up ◽  
Global Solutions ◽  
Initial Energy ◽  
Nonlinear Evolution Equations ◽  
Material Behavior ◽  
Memory Term ◽  
Viscoelastic Wave ◽  
Deformable Bodies

<p style='text-indent:20px;'>We consider a class of nonlinear evolution equations of second order in time, linearly damped and with a memory term. Particular cases are viscoelastic wave, Kirchhoff and Petrovsky equations. They appear in the description of the motion of deformable bodies with viscoelastic material behavior. Several articles have studied the nonexistence of global solutions of these equations due to blow-up. Most of them have considered non-positive and small positive values of the initial energy and recently some authors have analyzed these equations for any positive value of the initial energy. Within an abstract functional framework we analyze this problem and we improve the results in the literature. To this end, a new positive invariance set is introduced.</p>

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