Wave-breaking phenomena and global existence for the weakly dissipative generalized Novikov equation

In this paper, we mainly study several problems on the weakly dissipative generalized Novikov equation. We first establish the local well-posedness of solutions. We then give the precise blow-up scenarios for the generalized Novikov equation provided the momentum density associated with their initial data changes sign, and obtain the blow-up rate of blow-up solutions. Finally, we prove that the equation has a global solution provided the momentum density associated with their initial data do not change sign.

On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

<p style='text-indent:20px;'>We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>. Firstly we study the existence and stability of the maximal ground state <inline-formula><tex-math id="M3">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M4">\begin{document}$ N = N_c $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ N_c $\end{document}</tex-math></inline-formula> is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states <inline-formula><tex-math id="M6">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \beta\rightarrow 0^+ $\end{document}</tex-math></inline-formula>, and the optimal blow-up rate with respect to <inline-formula><tex-math id="M8">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> will be calculated.</p>

A nonlinear diffusion equation with reaction localized in the half-line

<abstract><p>We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = (u^m)_{xx}+a(x) u^p, $\end{document} </tex-math></disp-formula></p> <p>$ m, p &gt; 0 $ and $ a(x) = 1 $ for $ x &gt; 0 $, $ a(x) = 0 $ for $ x &lt; 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p &gt; 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p &gt; m $ or $ p = 1\neq m $.</p></abstract>

Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system

<p style='text-indent:20px;'>The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), &amp; \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &amp;{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &amp;{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &amp;{\rm in} \ \overline{\Omega}. \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a spatial bounded region in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{n} \ (n\geq2) $\end{document}</tex-math></inline-formula> and the boundary <inline-formula><tex-math id="M3">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula> of the spatial region <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is smooth. We give a sufficient condition to guarantee that the positive solution <inline-formula><tex-math id="M5">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> of the above problem must be a blow-up solution with a finite blow-up time <inline-formula><tex-math id="M6">\begin{document}$ t^* $\end{document}</tex-math></inline-formula>. In addition, an upper bound on <inline-formula><tex-math id="M7">\begin{document}$ t^* $\end{document}</tex-math></inline-formula> and an upper estimate of the blow-up rate on <inline-formula><tex-math id="M8">\begin{document}$ (u,v) $\end{document}</tex-math></inline-formula> are obtained.</p>

On the numerical solutions for a parabolic system with blow-up

<abstract><p>We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow-up rate and blow-up in $ L^\sigma $-norm. Moreover, the relation between blow-up of the exact solution and that of the numerical solution is also analyzed and discussed.</p></abstract>

Improving a method of constructing finite time blow-up solutions and its an application

We in this paper improve a method of establishing the existence of finite time blow-up solutions, and then apply it to study the finite time blow-up, the blow-up time and the blow-up rate of the weak solutions on the initial boundary problem of u_t - \Delta u_{t} - \Delta u_{t} = |u|^{p - 1}u. By applying this improved method, we prove that I(u_{0}) < 0 is a sufficient condition of the existence of the finite time blow-up solutions and \frac{2(p - 1)^{-1}\|u_{0}\|_{H_{0}^{1}}^{2}}{(p - 1) \|\nabla u_{0}\|_{2}^{2} - 2(p + 1)J(u_{0})} is an upper bound for the blow-up time, which generalize the blow-up results of the predecessors in the sense of the variation. Moreover, we estimate the upper blow-up rate of the blow-up solutions, too.

Type II singularities on complete non-compact Yamabe flow

This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

Some Results on Blow-up Phenomenon for Nonlinear Porous Medium Equations with Weighted Source

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut −4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.

Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in Rn.