Stochastic Parabolic Equations in the Whole Space

2014 ◽  
pp. 121-144
Keyword(s):  
Parabolic Equations ◽  
Whole Space

scholarly journals Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

2021 ◽  
Author(s):  
Nguyen Duc Phuong ◽  
Nguyen Anh Tuan ◽  
Devendra Kumar ◽  
Nguyen Huy Tuan
Keyword(s):  
Mild Solution ◽  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Nonlinear Memory ◽  
Main Equation ◽  
Memory Source ◽  
Whole Space ◽  
Initial Boundary Value

In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace  of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefler function, we will give some results on the existence and the uniqueness of mild solution for the Problem \eqref{Main-Equation} below. When $ \nu=1 $, we will introduce some $ L^p-L^q $ estimates, and base on them we derive the global existence of a mild solution in the whole space $ \R^d. $

scholarly journals Liouville Theorems for Fractional Parabolic Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenxiong Chen ◽  
Leyun Wu
Keyword(s):  
Half Space ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Qualitative Properties ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Narrow Region ◽  
Liouville Type Theorems ◽  
Whole Space

Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.

Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations

2017 ◽  
Vol 147 (6) ◽  
pp. 1311-1331 ◽  
Author(s):  
Xiaoli Zhu ◽  
Fuyi Li ◽  
Yuhua Li
Keyword(s):  
Weak Solution ◽  
Global Existence ◽  
Parabolic Equations ◽  
Integral Form ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Necessary Condition ◽  
Blowup Of Solutions ◽  
Decay Behaviour ◽  
Whole Space

In this paper we are interested in a sharp result about the global existence and blowup of solutions to a class of pseudo-parabolic equations. First, we represent a unique local weak solution in a new integral form that does not depend on any semigroup. Second, with the help of the Nehari manifold related to the stationary equation, we separate the whole space into two components S+ and S– via a new method, then a sufficient and necessary condition under which the weak solution blows up is established, that is, a weak solution blows up at a finite time if and only if the initial data belongs to S–. Furthermore, we study the decay behaviour of both the solution and the energy functional, and the decay ratios are given specifically.

scholarly journals Uniqueness of limit flow for a class of quasi-linear parabolic equations

2017 ◽  
Vol 6 (2) ◽  
pp. 243-276 ◽  
Author(s):  
Marco Squassina ◽  
Tatsuya Watanabe
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Global Solutions ◽  
Content Type ◽  
Linear Parabolic Equations ◽  
Whole Space ◽  
Characterization Of Solutions ◽  
Entire Trajectory ◽  
Relevant Class

AbstractWe investigate the issue of uniqueness of the limit flow for a relevant class of quasi-linear parabolic equations defined on the whole space. More precisely, we shall investigate conditions which guarantee that the global solutions decay at infinity uniformly in time and their entire trajectory approaches a single steady state as time goes to infinity. Finally, we obtain a characterization of solutions which blow up, vanish or converge to a stationary state for initial data of the form ${\lambda\varphi_{0}}$ while ${\lambda>0}$ crosses a bifurcation value ${\lambda_{0}}$.

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