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Register a new userMoore-Gibson-Thompson equation with memory, part I: exponential decay of energy
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We are interested in the Moore-Gibson-Thompson(MGT) equation with memory begin{equation} onumber tau u_{ttt}+ alpha u_{tt}+c^2A u+bA u_t -int_0^tg(t-s)A w(s)ds=0. end{equation} We first classify the memory into three types. Then we study how a memory term creates damping mechanism and how the memory causes energy decay.
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We study a temporally third order (Moore-Gibson-Thompson) equation with a memory term. Previously it is known that, in non-critical regime, the global solutions exist and the energy functionals decay to zero. More precisely, it is known that the energy has exponential decay if the memory kernel decays exponentially. The current work is a generalization of the previous one (Part I) in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does, exponentially or not.
We consider the MGT equation with memory $$partial_{ttt} u + alpha partial_{tt} u - beta Delta partial_{t} u - gammaDelta u + int_{0}^{t}g(s) Delta u(t-s) ds = 0.$$ We prove an existence and uniqueness result removing the convexity assumption on the convolution kernel $g$, usually adopted in the literature. In the subcritical case $alphabeta>gamma$, we establish the exponential decay of the energy, without leaning on the classical differential inequality involving $g$ and its derivative $g$, namely, $$g+delta gleq 0,quaddelta>0,$$ but only asking that $g$ vanishes exponentially fast.
In this paper we consider the initial value {problem $partial_{t} u- Delta u=f(u),$ $u(0)=u_0in exp,L^p(mathbb{R}^N),$} where $p>1$ and $f : mathbb{R}tomathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|sim mbox{e}^{|u|^q}$ as $|u|to infty$,} $|f(u)|sim |u|^{m}$ as $uto 0,$ $0<qleq pleq,m,;{N(m-1)over 2}geq p>1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$
Let $ngeq 3$, $alpha$, $betainmathbb{R}$, and let $v$ be a solution $Delta v+alpha e^v+beta xcdot abla e^v=0$ in $mathbb{R}^n$, which satisfies the conditions $lim_{Rtoinfty}frac{1}{log R}int_{1}^{R}rho^{1-n} (int_{B_{rho}}e^v,dx)drhoin (0,infty)$ and $|x|^2e^{v(x)}le A_1$ in $R^n$. We prove that $frac{v(x)}{log |x|}to -2$ as $|x|toinfty$ and $alpha>2beta$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.
In this paper we consider the problem: $partial_{t} u- Delta u=f(u),; u(0)=u_0in exp L^p(R^N),$ where $p>1$ and $f : RtoR$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $exp L^p_0(R^N)$ for $f(u)sim mbox{e}^{|u|^q},;0<qleq p,; |u|to infty.$ However, if for some $lambda>0,$ $displaystyleliminf_{sto infty}left(f(s),{rm{e}}^{-lambda s^p}right)>0,$ then non-existence occurs in $exp L^p(R^N).$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|sim |u|^{m}$ as $uto 0,$ ${N(m-1)over 2}geq p$, we show that the solution is global. In particular, $p-1>0$ sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m$.
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