Related papers: Complex-plane singularity dynamics for blow up in …
We present a study by computer simulations of a class of complex-valued solutions of the three-dimensional Navier-Stokes equations in the whole space, which, according to Li and Sinai, present a blow-up (singularity) at a finite time. The…
In this paper we study a system of delay differential equations from the viewpoint of a finite time blow-up of the solution. We prove that the system admits a blow-up solution, no matter how small the length of the delay is. In the…
In this paper, we investigate the computational complexity of solutions to the Laplace and the diffusion equation. We show that for a certain class of initial-boundary value problems of the Laplace and the diffusion equation, the solution…
We study, under the radial symmetry assumption, the solutions to the fractional Schr\"odinger equations of critical nonlinearity in $\mathbb R^{1+d}, d \geq 2$, with L\'{e}vy index ${2d}/({2d-1}) < \al < 2$. We firstly prove the linear…
We consider the nonlinear heat equation with a nonlinear gradient term: $\partial_t u =\Delta u+\mu|\nabla u|^q+|u|^{p-1}u,\; \mu>0,\; q=2p/(p+1),\; p>3,\; t\in (0,T),\; x\in \R^N.$ We construct a solution which blows up in finite time…
We study the asymptotic behavior of complex discrete evolution equations of Ginzburg- Landau type. Depending on the nonlinearity and the data of the problem, we find different dynamical behavior ranging from global existence of solutions…
We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…
In this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite…
We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal…
In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a…
The conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and the conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations are…
In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions for a class of logistic type quasilinear problem in a smooth bounded domain involving the $p$-Laplacian operator, where the…
We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\Bbb R^2$ into a smooth compact revolution surface of $\Bbb R^3$ which reduces to the semilinear parabolic problem $$\partial_t u…
The behaviour of solutions for a non-linear diffusion problem is studied. A subordination principle is applied to obtain the variation of parameters formula in the sense of Volterra equations, which leads to the integral representation of a…
This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where…
We study long time dynamics of non-radial solutions to the focusing inhomogeneous nonlinear Schr\"odinger equation. By using the concentration/compactness and rigidity method, we establish a scattering criterion for non-radial solutions to…
This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing…
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 study the blow-up question for the diffusion equation involving a nonlocal derivative in time defined by convolution with a nonnegative and nonincreasing kernel, and a nonlocal operator in space driven by a nonnegative radial L\'evy…
We are concerned with the asymptotic analysis of positive blow-up boundary solutions for a class of quasilinear elliptic equations with absorption term. By means of the Karamata theory we establish the first two terms in the expansion of…