相关论文: Blow-up solutions for a Kirchhoff type elliptic eq…
We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^{\alpha}(x,\partial{}B)\Phi_p(\nabla u)) = b(x)f(u)$ in a ball B,…
It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is…
We prove the existence of energy solutions of the energy critical focusing wave equation in R^3 which blow up exactly at x=t=0. They decompose into a bulk term plus radiation term. The bulk is a rescaled version of the stationary "soliton"…
In the present paper, we investigate blow-up and lifespan estimates for a class of semilinear hyperbolic coupled system in $\mathbb{R}^n$ with $n\geqslant 1$, which is part of the so-called Nakao's type problem weakly coupled a semilinear…
The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps $\mathbb R^{d}\to S^{d}$, restricted to…
In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schr\"odinger equation (NLS), $$ \textnormal{i} \, \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \R…
We consider a question posed in [GSZZ22], namely the blow-up of the PDE $$u_t + (b(t,x) u^{1+k})_x = u_{xx}$$ when $b$ is uniformly bounded, Lipschitz and $k = 2$. We give a complete answer to the behavior of solutions when $b$ belongs to…
We prove a blow-up criterion for the solutions to the $\nu$-dimensional Patlak-Keller-Segel equation in the whole space. The condition is new in dimension three and higher. In dimension two it is exactly Dolbeault's and Perthame's blow-up…
We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms.…
In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small…
The blowup is studied for the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative…
In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)\Delta u = \lambda u +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u…
We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of…
This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type. By means of blow-up analysis techniques, we prove an a-priori estimate in a weighted space of continuous functions. From this…
This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and…
We consider a nonlocal parabolic PDE, which may be regarded as the standard semilinear heat equation with power nonlinearity, where the nonlinear term is divided by some Sobolev norm of the solution. In this paper, we are interested in…
We study finite time blow-up and global existence of solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term. We show that for small enough initial data, if…
In this paper, we study the existence of normalized solutions to the following Kirchhoff equation with a perturbation: $$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda…
In this article, we study the log-log blowup dynamics for the mass critical nonlinear Schr\"odinger equation on $\mathbb{R}^{2}$ under rough but structured random perturbations at $L^{2}(\mathbb{R}^2)$ regularity. In particular, by…
In this paper, we establish blow-up rates for higher-order semilinear parabolic equations with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear…