Related papers: Blow-up solutions for a Kirchhoff type elliptic eq…
We consider the nonlinear Schr\"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same…
In this paper, we investigate the existence of normalized solutions for the following nonlinear Kirchhoff type problem \begin{equation*} \begin{cases} -(a+b\int_{\Omega}\vert\nabla u\vert^2dx)\Delta u+\lambda u=\vert u\vert^{p-2}u & \text{…
We identify the blow-up rate of a solution to a generalised Blasius equation, that we came across while studying a probabilistic model of "Poissonian burning" in Euclidean space. Our proof involves the study of the long-time behaviour of…
In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schr\"{o}dinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type.…
In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} -\Delta u + V(x) u = \mu_q u + a|u|^q u \quad {\rm in}\quad…
In this paper we consider self-similar blow-up solutions for the generalized deterministic KPZ~equation $u_t = u_{xx} + \lambda \vert u_x \vert ^q, \lambda > 0, q > 2.$ The asymptotic behavior of self-similar solutions are studied.
This paper is concerned with blow-up solutions of the 4-dimensional energy critical heat equation $u_t=\Delta u + u^3$. Our main result is to show that the existence of type II blowup solutions, and…
Under some conditions we give a blow-up analysis for solutions of an equation with Dirichlet boundary condition.
An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.
Following our previous paper in the radial case, we consider blow-up type II solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the…
In this paper, we consider the initial boundary value problem for a pseudo-parabolic Kirchhoff equation with logarithmic nonlinearity. We use the potential well method to give a threshold result of global existence and finite-time blow-up…
We consider the $L^2$-supercritical nonlinear Schr\"{o}dinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the…
This paper is dedicated to the blow-up solution for the divergence Schr\"{o}dinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where…
We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation $u_{tt}-u_{xx} + u - |u|^{2\alpha} u =0$ on the line. Using mixed numerical and analytical methods we find that solutions starting from even…
In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying…
We give a blow-up behavior for solutions to a problem with singularity and with Dirichlet condition. An application, we have a compactness of the solutions to this Problem with singularity and Lipschitz conditions.
In this paper we develop a blow up theory for the parabolic-elliptic Keller-Segel system, which can be viewed as a parabolic counterpart to the Liouville equation. This theory is applied to the study of first time singularities, ancient…
We study blowup solutions of the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\R^n\times(0,T)$. A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas, Herrero and Vel\'azquez…
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there…
In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as…