scholarly journals Randomization improved Strichartz estimates and global well-posedness for supercritical data

2021 ◽  
pp. 1-33
Author(s):  
Nicolas Burq ◽  
Joachim Krieger
Keyword(s):  
Strichartz Estimates ◽  
Well Posedness

scholarly journals Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bixiang Wang
Keyword(s):  
Asymptotic Behavior ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Colored Noise ◽  
Unbounded Domains ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Natural Energy ◽  
Random Attractors ◽  
Supercritical Nonlinearity

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ n\le 6 $\end{document}</tex-math></inline-formula>. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.</p>

scholarly journals Local in Time Strichartz Estimates for the Dirac Equation on Spherically Symmetric Spaces

2020 ◽  
Author(s):  
Federico Cacciafesta ◽  
Anne-Sophie de Suzzoni
Keyword(s):  
Dirac Equation ◽  
Symmetric Spaces ◽  
Nonlinear Models ◽  
Strichartz Estimates ◽  
Spherically Symmetric ◽  
Well Posedness

Abstract We prove local in time Strichartz estimates for the Dirac equation on spherically symmetric manifolds. As an application, we give a result of local well-posedness for some nonlinear models.

scholarly journals Global well-posedness and attractors for the hyperbolic Cahn–Hilliard–Oono equation in the whole space

2016 ◽  
Vol 26 (07) ◽  
pp. 1357-1384 ◽  
Author(s):  
Anton Savostianov ◽  
Sergey Zelik
Keyword(s):  
Growth Rate ◽  
Schrödinger Equation ◽  
Global Attractor ◽  
Schrodinger Equation ◽  
Energy Space ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Whole Space

We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn–Hilliard–Oono equation in the whole space [Formula: see text] with the nonlinearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Schrödinger equation in [Formula: see text].

scholarly journals STRICHARTZ ESTIMATES FOR WAVE EQUATION WITH INVERSE SQUARE POTENTIAL

2013 ◽  
Vol 15 (06) ◽  
pp. 1350026 ◽  
Author(s):  
CHANGXING MIAO ◽  
JUNYONG ZHANG ◽  
JIQIANG ZHENG
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Strichartz Estimates ◽  
Linear Wave ◽  
Well Posedness ◽  
Linear Wave Equation ◽  
Admissible Pairs

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential [Formula: see text] for power p being in some regime when the initial data are radial. This result extends the well-posedness result in Planchon, Stalker, and Tahvildar-Zadeh.

scholarly journals Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zijun Chen ◽  
Shengkun Wu
Keyword(s):  
Normal Form ◽  
Unique Solution ◽  
Strichartz Estimates ◽  
Well Posedness ◽  
Zakharov System ◽  
Initial Values

<p style='text-indent:20px;'>The Zakharov system in dimension <inline-formula><tex-math id="M2">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula> is shown to have a local unique solution for any initial values in the space <inline-formula><tex-math id="M3">\begin{document}$ H^{s} \times H^{l} \times H^{l-1} $\end{document}</tex-math></inline-formula>, where a new range of regularity <inline-formula><tex-math id="M4">\begin{document}$ (s, l) $\end{document}</tex-math></inline-formula> is given, especially at the line <inline-formula><tex-math id="M5">\begin{document}$ s-l = -1 $\end{document}</tex-math></inline-formula>. The result is obtained mainly by the normal form reduction and the Strichartz estimates.</p>

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

WELL-POSEDNESS AND NUMERICAL SOLUTION OF A NONLINEAR VOLTERRA PARTIAL INTEGRO-DIFFERENTIAL EQUATION MODELING A SWELLING POROUS MATERIAL

2014 ◽  
Vol 17 (9) ◽  
pp. 763-784 ◽  
Author(s):  
Keith J. Wojciechowski ◽  
Jinhai Chen ◽  
Lynn Schreyer-Bennethum ◽  
Kristian Sandberg
Keyword(s):  
Differential Equation ◽  
Porous Material ◽  
Numerical Solution ◽  
Equation Modeling ◽  
Well Posedness ◽  
Integro Differential Equation ◽  
Differential Equation Modeling

scholarly journals The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

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