scholarly journals On a quasilinear nonlocal Benney system

2017 ◽  
Vol 14 (01) ◽  
pp. 135-156 ◽  
Author(s):  
João-Paulo Dias ◽  
Filipe Oliveira
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Finite Time ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Bound State ◽  
Global Weak Solutions

We study a quasilinear nonlocal Benney system and establish the existence and uniqueness of strong local in time solutions to the corresponding Cauchy problem. We also show, under certain conditions, the blow-up of such solutions in finite time. Furthermore, we prove the existence of global weak solutions and exhibit bound-state solutions to this system.

scholarly journals Initial and Boundary Value Problems for a Class of Nonlinear Metaparabolic Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Huafei Di ◽  
Zefang Song
Keyword(s):  
Energy Level ◽  
Boundary Value Problems ◽  
Weak Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Boundary Value ◽  
Galerkin Approximation ◽  
Initial Energy ◽  
Potential Well ◽  
Global Weak Solutions

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equations u t − β u x x − k u x x t + γ u x x x x = f u x x . At low initial energy level ( J u 0 < d ), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.

scholarly journals Global weak solutions to the generalized mCH equation via characteristics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fanqin Zeng ◽  
Yu Gao ◽  
Xiaoping Xue
Keyword(s):  
Weak Solutions ◽  
Radon Measure ◽  
Measure Space ◽  
Blow Up ◽  
Space Time ◽  
Classical Solutions ◽  
Lagrangian Dynamics ◽  
Global Weak Solutions ◽  
Compactness Arguments ◽  
Mollification Method

<p style='text-indent:20px;'>In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns <inline-formula><tex-math id="M1">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> into its Lagrangian dynamics for characteristics <inline-formula><tex-math id="M2">\begin{document}$ X(\xi,t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \xi\in\mathbb{R} $\end{document}</tex-math></inline-formula> is the Lagrangian label. When <inline-formula><tex-math id="M4">\begin{document}$ X_\xi(\xi,t)&gt;0 $\end{document}</tex-math></inline-formula>, we use the solutions to the Lagrangian dynamics to recover the classical solutions with <inline-formula><tex-math id="M5">\begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M6">\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}</tex-math></inline-formula>) to the gmCH equation. The classical solutions <inline-formula><tex-math id="M7">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> to the gmCH equation will blow up if <inline-formula><tex-math id="M8">\begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ T_{\max}&gt;0 $\end{document}</tex-math></inline-formula>. After the blow-up time <inline-formula><tex-math id="M10">\begin{document}$ T_{\max} $\end{document}</tex-math></inline-formula>, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula> in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.</p>

The critical exponents of parabolic equations and blow-up in Rn

1998 ◽  
Vol 128 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Yuan-wei Qi
Keyword(s):  
Cauchy Problem ◽  
Global Solution ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Parabolic Equations ◽  
Finite Time ◽  
Blow Up ◽  
Initial Value ◽  
The Cauchy Problem

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for p ≦ pc, where pc ≡ m + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

scholarly journals TheH1(R)Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Zhaowei Sheng ◽  
Shaoyong Lai ◽  
Yuan Ma ◽  
Xuanjun Luo
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Space Variable ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Dissipative Term ◽  
First Order ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Derivatives Of

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞)×R)∩L∞([0,∞);H1(R))provided that its initial valueu0(x)belongs to the spaceH1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

scholarly journals The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jae-Myoung Kim
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Global Existence ◽  
Viscous Fluid ◽  
Power Law ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Fluid Flows ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

scholarly journals Critical exponents for a semilinear parabolic equation with variable reaction

2012 ◽  
Vol 142 (5) ◽  
pp. 1027-1042 ◽  
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
M. Pérez-LLanos ◽  
J. D. Rossi
Keyword(s):  
Critical Exponents ◽  
Finite Time ◽  
Existence And Uniqueness ◽  
Parabolic Problem ◽  
Blow Up ◽  
Bounded Function ◽  
Semilinear Parabolic Equation ◽  
All Solutions ◽  
Image Position ◽  
Semilinear Parabolic

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem:where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1.When Ω = ℝN we show that if p− > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p− ≤ p+ ≤ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p− < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions.When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).

Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem

1995 ◽  
Vol 125 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Pierangelo Marcati ◽  
Roberto Natalini
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence Theorem ◽  
Weak Solutions ◽  
Hydrodynamic Model ◽  
Fractional Step ◽  
Large Initial Data ◽  
Global Weak Solutions ◽  
The Cauchy Problem

We investigate the Cauchy problem for a hydrodynamic model for semiconductors. An existence theorem of global weak solutions with large initial data is obtained by using the fractional step Lax—Friedrichs scheme and Godounov scheme.

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