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Register a new userGlobal existence and decay estimates for the heat equation with exponential nonlinearity
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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 $m.$
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In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $partial_{t} u+ Delta^2 u=f(u),;t>0,;xinR^N,$ with $f(u)sim mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)sim u^m$ as $uto 0,$ $m$ integer and $N(m-1)/4geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
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 $m$.
We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $ipartial_t u + (-Delta)^{frac{alpha}2} u = 2gammabeta e^{beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $mathcal{M}$, for a dispersion parameter $alpha>d$, some coupling constant $beta>0$, and $gamma eq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $gamma>0$, the measure is well-defined in the whole regime $alpha>d$ and $beta>0$ (Theorem 1.1 (i)), while in the focusing case $gamma<0$ its partition function is always infinite for any $alpha>d$ and $beta>0$, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure in the whole regime $alpha>d$ and $0<beta < beta^star_alpha$ for some natural parameter $0<beta^star_alphasim (alpha-d)$ (Theorem 1.3 (i)). In the large dispersion regime $alpha>2d$, we can improve this result by constructing a local deterministic flow for (expNLS) for any $beta>0$. Using the Gibbs measure, we prove that solutions are almost surely global for $0<beta llbeta^star_alpha$, and that the Gibbs measure is invariant (Theorem 1.3 (ii)). (iii) Finally, in the particular case $d=1$ and $mathcal{M}=mathbb{T}$, we are able to exploit some probabilistic multilinear smoothing effects to build a probabilistic flow for (expNLS) for $1+frac{sqrt{2}}2<alpha leq 2$, locally for arbitrary $beta>0$ and globally for $0<beta ll beta^star_alpha$ (Theorem 1.5).
The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^infty (mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power.
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
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