Special Session 41: Global and Blowup Solutions for Nonlinear Evolution Equations

Blow-up solutions for nonlinear parabolic equations

Shaohua Chen Cape Breton University Canada Co-Author(s):

Abstract: In this talk, we will present a blow-up problem for nonlinear parabolic equations. We introduce a new method to improve the existing results and present some numerical results to support theoretical results.

High energy blowup and blowup time for a class of semilinear parabolic equations with singular potential on manifolds with conical singularities

Yuxuan Chen Heilongjiang University Peoples Rep of China Co-Author(s):    Vicentiu D. Radulescu, Runzhang Xu

Abstract: We consider a class of semilinear parabolic equations with singular potential on manifolds with conical singularities. At high initial energy level $J(u_0)>d$, we present a new sufficient condition to describe the global existence and nonexistence of solutions, respectively. Moreover, by applying the Levine`s concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy $J(u_0)>0$, including the upper bound of blowup time. Finally, we show a lower bound of the blowup time and blowup rate under arbitrary initial energy level.

Global non-existence of a coupled parabolic-hyperbolic system of thermoelastic type with history

Jorge A Esquivel-Avila Universidad Autonoma Metropolitana Mexico Co-Author(s):

Abstract: We consider two abstract systems of parabolic-hyperbolic type that model thermoelastic problems. We study the influence of the physical constants and the initial data on the nonexistence of global solutions that in our framework are produced by the blow-up in finite time of the norm of the solution in the phase space. We employ a differential inequality to find sufficient conditions that produce the blow-up. To that end, we construct a set that is positive invariant for any positive value of the initial energy. As a result we found that the coupling with the parabolic equation stabilizes the system, as well as the damping term in the hyperbolic equation. Moreover, for any pair of positive values $(\xi,\epsilon)$, there exist initial data such that the corresponding solution with initial energy $\xi$ blows-up at a finite time less than $\epsilon$. Our purpose is to improve results previously published in the literature.

Nonlinear stability of shock profiles to Burgers equation with critical fast diffusion and singularity

Jingyu Li Northeast Normal University Peoples Rep of China Co-Author(s):    Xiaowen Li, Ming Mei, Jean-Christophe Nave

Abstract: We are interested in the Burgers` equation featuring critical fast diffusion in form of $u_t+f(u)_x = (\ln u)_{xx}$. The solution possesses a strong singularity when $u=0$ hence bringing technical challenges. We investigate the asymptotic stability of viscous shocks, particularly those with shock profiles vanishing at the far field $x=+\infty$. To overcome the singularity, we introduce some weight functions and show the nonlinear stability of shock profiles through the weighted energy method.

Longtime dynamics for a class of strongly damped wave equations with variable exponent nonlinearities

Yanan Li Harbin Engineering University Peoples Rep of China Co-Author(s):    Yamei Li; Zhijian Yang

Abstract: This talk considers the global well-posedness and the longtime dynamics for a class of strongly damped wave equations with evolutional $p(x,t)$-Laplacian, $q(x,t)$-growth source term, and the perturbed parameter $\lambda\in [0,1]$. By using the monotone operator technique and the quasi-stability method, we show the global well-posedness of weak solutions in the time-dependent phase space $[W_0^{1, p(\cdot, t)}(\Omega)\cap L^{q(\cdot, t)}(\Omega)]\times L^2(\Omega)$ and the existence of pullback $\mathscr D$-attractor and pullback $\mathscr D$-exponential attractor, which are also continuous concerning the parameter $\lambda$. This work also establishes the additional regularity of weak solutions and attractors in the semilinear case, which extends the analysis and the results for these types of models with constant exponent nonlinearities.

Qualitative properties of solution for a class of heat equations

Junmiao Liu Harbin Engineering University Peoples Rep of China Co-Author(s):    Yanbing Yang

Abstract: This paper delves into the initial-boundary value problem of the ``Euclidean Bosonic Heat Equation``, conducting a comprehensive analysis within the framework of potential wells. In this paper, the asymptotic behavior of the solution is significantly improved from the polynomial form to the stronger exponential decay form. Furthermore, in the range of parameters, the finite time blowup of the solution at any high energy level is proved.

Sobolev Anisotropic inequalities with monomial weights

Maria Rosaria M Posteraro Universita di Napoli Federico II Italy Co-Author(s):    Feo F., Martin J., Passarelli di Napoli A.

Abstract: I will present some anisotropic Sobolev inequalities in $\mathbb{R}^{n}$ with a monomial weight in the general setting of rearrangement invariant spaces (e.g. $L^{p}$, Lorentz, Orlicz, etc...). The monomial weights are defined by $\begin{equation*} d\mu (x):=x^{A}dx=|x_{1}|^{A_{1}}\cdots |x_{n}|^{A_{n}}dx, \label{mu} \end{equation*}$ where $A=(A_{1},A_{2},\dots ,A_{n})$ is a vector in $\mathbb{R}^{n}$ with $% A_{i}\geq 0$ for $i=1,\dots ,n$. Some applications to study local boundedness of minimizers of a class of non uniformly elliptic integral functionals are also given.

Global existence for aggregation-diffusion systems with irregular kernels

Yurij Salmaniw University of Oxford England Co-Author(s):    J. Carrillo; J. Skrzeczkowski

Abstract: Aggregation-diffusion equations and systems have grow rapidly in their population. Models featuring nonlocal interactions through spatial convolution have been applied to several areas, including the physical, chemical, and biological sciences. A typical strategy to establish well-posedness is to use regularity properties of the kernels themselves; however, for many model problems such regularity is not available. One such example is the top-hat kernel which is discontinuous. In this talk, I will present recent progress in establishing a robust well-posedness theory for a class of nonlocal aggregation-diffusion models with minimal regularity requirements on the interaction kernel in any spatial dimension on either the whole space or the torus. Starting with the scalar equation, we first establish the existence of a global weak solution in a small mass regime for merely bounded kernels. Under some additional hypotheses, we show the existence of a global weak solution for any initial mass. In typical cases of interest, these solutions are unique and classical. I will then discuss the generalisation to the n-species system for the regimes of small mass and arbitrary mass. We will conclude with some consequences of these theorems for several models typically found in ecological applications. This is joint work with Dr. Skrzeczkowski and Prof. Jose Carrillo.

Global quantitative stability of wave equations with strong and weak dampings

Runzhang Xu Harbin Engineering University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are concerned with the description of global quantitative stability of wave equations with linear strong damping and linear or nonlinear weak damping. By giving some energy decay estimates, we obtain several conclusions about the continuous dependence of the global solution on the initial data and the coefficients of the strong damping term and linear or nonlinear weak damping term. This work also establishes a new idea to use the dissipative effect to obtain the better continuous dependence conclusions, which also reflect the dissipative properties of the solution.

Recent progresses on stochastic Zakharov systems

Deng Zhang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Deng Zhang

Abstract: In this talk we review very recent progresses on stochastic Zakharov systems in dimensions three and four. The Zakharov system couples Schroedinger and wave equations, and reaches the energy criticality in dimension four. We will mainly show the global well-posedness below the ground state and the noise regularization effects on blow-up and scattering dynamics. This talk is based on joint works with Sebastian Herr, Michael Roeckner and Martin Spitz.