Related papers: On solutions to $-\Delta u = V u$ near infinity

We consider convex potentials $W:\R\to [0,\infty)$ vanishing at $0$ and growing sufficiently fast at $\pm\infty$. Given any open set $\Omega\subset\R^n$ with Lipschitz and compact boundary, we prove the existence and uniqueness of a…

We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - \Delta u = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous…

We establish the uniqueness of the positive solution for equations of the form $-\Delta u=au-b(x)f(u)$ in $\Omega$, $u|\_{\partial\Omega}=\infty$. The special feature is to consider nonlinearities $f$ whose variation at infinity is…

We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential…

We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma , \] where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the…

We are concerned with the qualitative analysis of positive singular solutions with blow-up boundary for a class of logistic-type equations with slow diffusion and variable potential. We establish the exact blow-up rate of solutions near the…

This paper investigates the initial boundary value problem for a fractional pseudo-parabolic equation with singular potential. The global existence and blow-up of solutions to the initial boundary value problem are obtained at low initial…

This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\frac{\mu}{1+t} \partial_t u$, the mass term $\frac{\nu^2}{(1+t)^2} u$, and a time-derivative nonlinearity $|…

We investigate the quantitative unique continuation property for solutions to $$\Delta^2_{X} u = V u,$$ where $\Delta_{X} = \Delta_{x} + |x|^{2\beta} \Delta_{y}$ ($0 < \beta \leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$,…

We study the regularity properties of a weak solution to the boundary value problem for the equation $-\Delta \rho +a u=f$ in a bounded domain $\Omega\subset \mathbb{R}^N$, where $\rho=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla…

We prove the existence of solutions for the singularly perturbed Schr\"odinger--Newton system {ll} \hbar^2 \Delta \psi - V(x) \psi + U \psi =0 \hbar^2 \Delta U + 4\pi \gamma |\psi|^2 =0 . \hbox{in $\mathbb{R}^3$} with an electric potential…

We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1\leq p<m$ and…

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 $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up…

We prove the unique continuation property for the differential inequality $|(-\Delta)^{\alpha/2}u|\leq|V(x)u|$, where $0<\alpha<n$ and $V\in L_{\textrm{loc}}^{n/\alpha,\infty}(\mathbb{R}^n)$, $n\geq3$.

In this paper we are interested in the blowup of a geometric constant $\mathfrak{C}(\delta)$ appearing in the optimal quantitative unique continuation property for wave operators. In a particular geometric context we prove an upper bound…

In this paper, we consider the following semi-linear complex heat equation \begin{eqnarray*} \partial_t u = \Delta u + u^p, u \in \mathbb{C} \end{eqnarray*} in $\mathbb{R}^n,$ with an arbitrary power $p,$ $ p > 1$. In particular, $p$ can be…

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…

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…

For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions,…

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…