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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.
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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 $m.$
Consider the Landau equation with Coulomb potential in a periodic box. We develop a new $L^{2}rightarrow L^{infty }$ framework to construct global unique solutions near Maxwellian with small $L^{infty } $norm. The first step is to establish global $L^{2}$ estimates with strong velocity weight and time decay, under the assumption of $L^{infty }$ bound, which is further controlled by such $L^{2}$ estimates via De Giorgis method cite{golse2016harnack} and cite{mouhot2015holder}. The second step is to employ estimates in $S_{p}$ spaces to control velocity derivatives to ensure uniqueness, which is based on Holder estimates via De Giorgis method cite{golse2016harnack}, cite{golse2015holder}, and cite{mouhot2015holder}.
We study the well-posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions. We give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3D under null condition, with small scaling invariant $dot{W}^{2,1}(mathbb{R}^3)$ data. We obtain a bilinear estimate that is effective to the hyperbolic systems which do not have any time decay. It allows us to achieve the boundedness of the weighted BV norm of the radial solution.
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 following non-local semi-linear parabolic equation with advection: for $1 le p<1+frac{2}{N}$, begin{equation*} begin{cases} u_t+v cdot abla u-Delta u=|u|^p-int_{mathbb T^N} |u|^p quad & textrm{on} quad mathbb T^N, u textrm{periodic} quad & textrm{on} quad partial mathbb T^N end{cases} end{equation*} with initial data $u_0$ defined on $mathbb T^N$. Here $v$ is an incompressible flow, and $mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbitrary data in $L^2$. We then study the global existence of the solutions under the following two scenarios: (1). when $v$ is a mixing flow; (2). when $v$ is a shear flow. More precisely, we show that under these assumptions, there exists a global solution to the above equation in the sense of $L^2$.
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