Related papers: Sharp non-explicit blow-up profile for perturbed n…
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is…
We consider radial decreasing solutions of the semilinear heat equation with exponential nonlinearity. We provide a relatively simple proof of the sharp upper estimates for the final blowup profile and for the refined space-time behavior.…
We consider the classical problem of the blowing-up of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blow-up point, and for each integer $k$, we construct a set of codimension $2k$ in…
We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u(x,t)$, the graph…
We consider the the pointwise estimates and the blow-up rate estimates for the zero Dirchilet problem of the semilinear heat equation with a gradient term.
This paper deals with the blow-up properties of positive solutions to a system of two heat equations.
We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions and we estimate the remainders.
Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition.…
We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for…
We consider in this paper a large class of perturbed semilinear wave equations with critical (in the conformal transform sense) power nonlinearity. We will show that the blow-up rate of any singular solution is given by the solution of the…
We consider in this paper a class of strongly perturbed semilinear wave equations with a non-characteristic point in one space dimension, for general initial data. Working in the framework of similarity variables, in \cite {MZ} Merle and…
In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a at blowup profile, already proved by Bricmont and Kupainen together with Herrero and Vel\'{a}zquez. Though our…
Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified…
We consider a vector-valued blow-up solution with values in $\mathbb{R}^m$ for the semilinear wave equation with power nonlinearity in one space dimension (this is a system of PDEs). We first characterize all the solutions of the associated…
We study the blowup behavior of a class of strongly perturbed wave equations with a focusing supercritical power nonlinearity in three spatial dimensions. We show that the ODE blowup profile of the unperturbed equation still describes the…
In this paper, the discretization of a nonlinear wave equation whose nonlinear term is a power function is introduced. The difference equation derived by discretizing the nonlinear wave equation has solutions which show characteristics…
In this paper we develop a blow up analysis for solutions of a planar semilinear elliptic equation involving exponential nonlinearities. Such solutions describe cosmic strings, and we show how their blow up behaviour is characterised by new…
We study the focusing semilinear heat equation with an additional defocusing H\'enon-type nonlinearity, the coupling of which is measured by a constant $c >0$. For $c \in (0,c^*)$, the model admits a closed-form self-similar blowup solution…
We consider positive radial decreasing blow-up solutions of the semilinear heat equation \begin{equation*} u_t-\Delta u=f(u):=e^{u}L(e^{u}),\quad x\in \Omega,\ t>0, \end{equation*} where $\Omega=\mathbb{R}^n$ or $\Omega=B_R$ and $L$ is a…
We consider in this paper a large class of perturbed semilinear wave equation with subconformal power nonlinearity. In particular, we allow log perturbations of the main source. We derive a Lyapunov functional in similarity variables and…