Related papers: Asymmetric blow-up for the SU(3) Toda System
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
We establish uniform a priori estimates for solutions of semilinear planar Hamiltonian elliptic systems in a ball with Dirichlet boundary conditions. We consider a broad class of coupled nonlinearities with asymptotic critical behaviour in…
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension…
We discuss the influence of possible spatial inhomogeneities in the coefficients of logistic source terms in parabolic-elliptic chemotaxis-growth systems of the form \begin{align*} u_t &= \Delta u - \nabla\cdot(u\nabla v) +…
We study positive blowing-up solutions of the system: $$u_{t}-\delta\Delta u=v^p,\,\,\, v_{t}-\Delta v=u^{q},$$ as well as of some more general systems. For any $p,\,q>1$, we prove single-point blow-up for any radially decreasing, positive…
We investigate soliton solutions of the Toda hierarchy on a quasi-periodic finite-gap background by means of the double commutation method and the inverse scattering transform. In particular, we compute the phase shift caused by a soliton…
Let $(M, g)$ be a compact Riemann surface with area $1$. We investigate the Toda system \begin{align} \begin{cases} -\Delta u_1 = 2\rho_1(h_1e^{u_1}-1) - \rho_2(h_2e^{u_2}-1),\\ -\Delta u_2 = 2\rho_2(h_2e^{u_2}-1) - \rho_1(h_1e^{u_1}-1),…
We prove a sufficient condition for the existence of a $T$-periodic solution for the planar system $\dot z=F(t,z)$, characterized by the growth to infinity of the rotations made in one period by solutions starting at increasingly large…
We analyze solutions of the Toda system and establish an optimal Moser-Trudinger inequality
It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \(\Omega \subset \mathbb{R}^2\): \begin{equation*} \begin{cases} u_t =…
For all non-symmetric discrete relativistic Toda type equations we establish a relation to 3D consistent systems of quad-equations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of $\mathbb…
This paper presents a novel approach to establish a blow-up mechanism for the forced 3D incompressible Euler equations, with a specific focus on non-axisymmetric solutions. We construct solutions on $\mathbb{R}^3$ within the function space…
This paper investigates the classification of solutions satisfying the polynomial energy growth condition near both the origin and infinity to the ${\mathrm SU}(n+1)$ Toda system on the punctured complex plane $\mathbb{C}^*$. The ${\mathrm…
In this paper we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. After introducing suitable functionals proposed by Lai-Takamura for the…
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…
This paper is motivated by the question of whether a sequence of solutions of a given integrable system can be blown up to obtain a solution of a different integrable system in the limit. We study a specific example of this phenomenon.…
We give a blow-up behavior for solutions to a problem with singularity and with Dirichlet condition. An application, we have a compactness of the solutions to this Problem with singularity and Lipschitz conditions.
In this paper we study blow-up phenomena in general coupled nonlinear Schrodinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several…
The combustion model is studied in three-dimensional (3D) smooth bounded domains with various types of boundary conditions. The global existence and uniqueness of strong solutions are obtained under the smallness of the gradient of initial…
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of…