Related papers: On Blowup for time-dependent generalized Hartree-F…
Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity…
A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order…
We study the behaviour of smooth solutions to the wave equation, $\square_g\psi=0$, in the interior of a fixed Schwarzschild black hole. In particular, we obtain a full asymptotic expansion for all solutions towards $r=0$ and show that it…
We show that the 1D Hou-Luo model on the real line admits exact self-similar finite-time blowup solutions with smooth self-similar profiles. The existence of these profiles is established via a fixed-point method that is purely analytic. We…
This paper is concerned with the two-species chemotaxis-competition model with degenerate diffusion, \[\begin{cases} u_t = \Delta u^{m_1} - \chi_1 \nabla\cdot(u\nabla w) + \mu_1 u (1-u-a_1v), &x\in\Omega,\ t>0,\\% v_t = \Delta v^{m_2} -…
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…
The possibility of finite-time, dispersive blow up for nonlinear equations of Schroedinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the…
In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i \beta) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u, \quad \text{where } \beta, \delta, \alpha \in \mathbb{R}. $$ The study focuses on investigating…
In this paper, we consider the dynamics of the solution to the mass critical focusing Hartree equation with inverse-square potential in the energy space $H^{1}(\mathbb{R}^d)$. The main difficulties are the equation is \emph{not}…
The $\lambda \phi^4$ model in a finite volume is studied within a non-gaussian Hartree-Fock approximation (tdHF) both at equilibrium and out of equilibrium, with particular attention to the structure of the ground state and of certain…
Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of…
A new energy-based stochastic extension of the Schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all…
We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method.…
In this paper, we consider the Schr\"odinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of $\R^{d}$ with Dirichlet boundary conditions. We prove that solutions with…
An extension of the time-dependent Hartree-Fock-Bogoliubov theory (ETDHFB) which includes higher-order effects such as screening of the pairing correlation is proposed. ETDHFB is applied to a fermion system trapped in a harmonic potential…
We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.
In this paper, we study the blow-up radial solution of fully parabolic system with higher dimensional two species Cauchy problem for some initial condition. In addition, we show that the set of positive radial functions in…
In this paper we present a method for extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimension,…
Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions,…
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) +…