相关论文: Blow-up behavior outside the origin for a semiline…
We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this…
We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blow-up solution with a characteristic point, we refine the blow-up behavior first derived by Merle and Zaag. We also refine the geometry of…
We consider the semilinear wave equation in higher dimensions with power nonlinearity in the super-conformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up…
We investigate semilinear wave-type equations that can be recast as wave equations with derivatives perturbed by zero-order terms. This framework covers several well-studied cases, including the scale-invariant wave equation. In this…
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on…
We consider $u(x,t)$, a solution of $\partial_tu = \Delta u + |u|^{p-1}u$ which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to \mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to be the blow-up…
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 consider the semilinear wave equation with focusing energy-critical nonlinearity in space dimension 5 with radial data. It is known that a solution $(u, \partial_t u)$ which blows up at $t = 0$ in a neighborhood (in the energy norm) of…
We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-\Delta u =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant,…
This is the first of two papers devoted to the study of the properties of the blow-up surface for the $N$ dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in…
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 study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we…
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…
In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \frac{\mu}{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under…
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…
We consider semilinear wave equations with focusing power nonlinearities in odd space dimensions $d \geq 5$. We prove that for every $p > \frac{d+3}{d-1}$ there exists an open set of radial initial data in $H^{\frac{d+1}{2}} \times…
In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known…
We consider non-local in time semilinear subdiffusion equations on a bounded domain, where the kernel in the integro-differential operator belongs to a large class, which covers many relevant cases from physics applications, in particular…
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work (Abdelhedi-Zaag JDE 2021), we constructed a blow-up solution for that…
We consider the semilinear wave equation $$\partial_t^2 u -\Delta u =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We…