Related papers: Blowup behaviour for the nonlinear Klein--Gordon e…
We consider the focusing mass-critical nonlinear Schr\"odinger equation and prove that blowup solutions to this equation with initial data in $H^s(\R^d)$, $s > s_0(d)$ and $d\geq 3$, concentrate at least the mass of the ground state at the…
We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes…
In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave…
A time-space fractional reaction-diffusion equation in a bounded domain is considered. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time. However, for realistic initial conditions,…
We study global behavior of radial solutions for the nonlinear wave equation with the focusing energy critical nonlinearity in three and five space dimensions. Assuming that the solution has energy at most slightly more than the ground…
We construct a finite time blow up solution for the nonlinear Schr\"odinger equation with the power nonlinearity whose coefficient is complex number. We generalize the range of both the power and the complex coefficient for the result of…
We consider the defocusing nonlinear wave equation $u_{tt}-\Delta u + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the…
We study the Klein-Gordon equation in one spatial and one temporal dimension. Physically, this equation describes the wave function of a relativistic spinless boson with positive rest mass. Mathematically, this is the most elementary…
We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schr\"odinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate…
Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In…
We study the 1D Klein-Gordon equation with variable coefficient nonlinearity. This problem exhibits an interesting resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the…
We consider dissipation in a recently proposed nonlinear Klein-Gordon dynamics that admits soliton-like solutions of the power-law form $e_q^{i(kx-wt)}$, involving the $q$-exponential function naturally arising within the nonextensive…
In this paper we consider the nonexistence of global solutions of a Klein-Gordon equation of the form \begin{eqnarray*} u_{tt}-\Delta u+m^2u=f(u)& (t,x)\in [0,T)\times\R^n. \end{eqnarray*} Here $m\neq 0$ and the nonlinear power $f(u)$…
We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schr\"{o}dinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete…
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first…
We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm…
We show global well-posedness in energy norm of the semi-relativistic Schr\"odinger-Poisson system of equations with attractive Coulomb interaction in ${\mathbb R}^3$ in the presence of pseudo-relativistic diffusion. We also discuss…
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…
In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the…
The numerical approximation of the semilinear Klein--Gordon equation in the $d$-dimensional space, with $d=1,2,3$, is studied by analyzing the consistency errors in approximating the solution. By discovering and utilizing a new cancellation…