Cauchy problem for a generalized weakly dissipative periodic two-component Camassa-Holm system
Electronic Journal of Differential Equations, 2014 In this article, we study a generalized weakly dissipative periodic two-component Camassa-Holm system. We show that this system can exhibit the wave-breaking phenomenon and determine the exact blow-up rate of strong solution to the system. In addition,
Wenxia Chen, Lixin Tian, Xiaoyan Deng
doaj
Existence and Destruction of Kantorovich Main Continuous Solutions of Nonlinear Integral Equations [PDF]
arXiv, 2013 The sufficient conditions are obtained for existence of the main solution of the nonlinear Volterra integral equation of the second kind on the semi-axis and on a finite interval. The method for computation of this boundary interval is designed. Beyond such integral the solution has the blow-up.
arxiv
Global dynamics for the generalised chemotaxis-Navier–Stokes system in $\mathbb{R}^3$
European Journal of Applied MathematicsWe consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$ : \begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla
Qingyou He, Ling-Yun Shou, Leyun Wu
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Hardy inequalities and non-explosion results for semigroups [PDF]
arXiv, 2014 We prove non-explosion results for Schr\"odinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.
arxiv
A critical non-homogeneous heat equation with weighted source
European Journal of Applied MathematicsSome qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source \begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end ...
Razvan Gabriel Iagar, Ariel Sánchez
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Life Span of Solutions for a Semilinear Heat Equation with Initial Data Non-Rarefied at $\infty$ [PDF]
arXiv, 2015 We study the Cauchy problem for a semilinear heat equation with initial data non-rarefied at $\infty$. Our interest lies in the discussion of the effect of the non-rarefied factors on the life span of solutions, and some sharp estimates on the life span is established.
arxiv
Damping to prevent the blow-up of the Korteweg-de Vries equation [PDF]
arXiv, 2015 We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on $\mathscr{L}_{\gamma}$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.
arxiv
A Symmetry problem for some quasi-linear equations in Euclidean space [PDF]
arXivWe prove sharp asymptotic estimates for the gradient of positive solutions to certain nonlinear $p$-Laplace equations in Euclidean space by showing symmetry and uniqueness of positive solutions to associated limiting problems.
arxiv
Classification of minimal mass blow-up solutions for an $L^2$ critical inhomogeneous NLS [PDF]
arXiv, 2015 We establish the classification of minimal mass blow-up solutions of the $L^2$ critical inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^{\frac{4-2b}{N}}u = 0, \] thereby extending the celebrated result of Merle from the classic case $b=0$ to the case $0
arxiv
Dichotomy Results for the Electromagnetic Schrödinger Equation [PDF]
arXivThe electromagnetic nonlinear Schr\"odinger (emNLS) equation is a variant of the well-studied nonlinear Schr\"odinger equation. In this article, we consider questions of global existence or blow-up for emNLS in dimensions 3 and higher.
arxiv