Related papers: The high exponent limit $p \to \infty$ for the one…
This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other…
We study the boundary of the liquid region $\mathcal{L}$ in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line $(x,1)$, $x\in\mathbb{R}\equiv…
For convex co-compact hyperbolic quotients $X=\Gamma\backslash\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the…
We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<\sigma<2$, $N\ge1$. If the…
This paper studies the upper bound of the lifespan of classical solutions of the initial value problems for one dimensional wave equations with quasilinear terms of space-, or time-derivatives of the unknown function. The results are same…
In this paper, we expand on results from our previous paper "The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples" [1] by showing that the failure of "peeling" (and, thus, of smooth null infinity) in a neighbourhood of…
We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities…
In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight $w$. Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting…
For $p \in (1, \infty),$ for an integer $N \geq 2$ and for a bounded Lipschitz domain $\Omega$, we consider the following nonlinear Steklov bifurcation problem \begin{equation*} \begin{aligned} -\Delta_p \phi & = 0 \; \text{in} \ \Omega, \\…
In this paper we study a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in $W^{1,\infty}$ in the space-time domain $(0,1)\times [0,+\infty)$. Then…
In this paper, we study the initial boundary value problem for the two dimensional strong damped wave equation with exponentially growing source and damping terms. We first show the well-posedness of this problem and then prove the…
We use the perturbation theory to build solitary wave solutions $\phi_\omega(x)e^{-i\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\bar\psi\psi)\beta\psi$, with…
We consider a nonlinear Schr\"odinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are…
We investigate an initial-boundary value problem for the quasilinear Westervelt equation which models the propagation of sound in fluidic media. We prove that, if the initial data are sufficiently small and regular, then there exists a…
We consider in this paper blow-up solutions of the semilinear wave equation in one space dimension, with an exponential source term. Assuming that initial data are in $H^{1}_{loc}\times L^2_{loc}$ or some times in $ W^{1,\infty}\times…
We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\Delta^2 \phi = \phi^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions…
In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the…
We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} \psi)^2 + \frac{ g_2^2}{2} ( {\bphi} \phi)^2 + g_3^2 ({\bpsi} \psi) ( {\bphi} \phi)$ as well as…
The nonlinear and nonlocal PDE $$ |v_t|^{p-2}v_t+(-\Delta_p)^sv=0 \, , $$ where $$ (-\Delta_p)^s v\, (x,t)=2 \,\text{PV} \int_{\mathbb{R}^n}\frac{|v(x,t)-v(x+y,t)|^{p-2}(v(x,t)-v(x+y,t))}{|y|^{n+sp}}\, dy, $$ has the interesting feature…
Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are…