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arXiv 1407.1969math.AP
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具有吸收项的Hamilton-Jacobi抛物型方程解的局部和整体估计

Local and global estimates of solutions of Hamilton-Jacobi parabolic equation with absorption

Marie-Françoise Bidaut-Véron

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摘要：

我们得到了方程\[ u_{t}-Δu+|\nabla u|^{q}=0 \]在$Q_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$中的非负解的新的先验估计，其中$q>0,$和$Ω=\mathbb{R}^{N},$或$Ω$是$\mathbb{R}^{N}$和$u=0$在$\partialΩ\times\left( 0,T\right) .$上的光滑有界区域。在$Ω=\mathbb{R}^{N},$的情况下，我们证明了方程的任何解$u\in C^{2,1}(Q_{\mathbb{R}^{N},T})$ 1.1在$Q_{\mathbb{R}^{N} ,T}$中，特别是任何弱解，如果$q\leqq2),$不带条件作为$\left\vert x\right\vert \rightarrow\infty,$满足普适估计\[ \left\vert \nabla u(.,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(.,t)}% {t},\qquad\text{in }Q_{\mathbb{R}^{N},T}. \]，并且我们证明了$u$的增长性是由$C(t+t^{-1/(q-1}% )(1+\left\vert x\right\vert ^{q^{\prime}}),$限制的，其中$C$依赖于$u.$.我们还给出了初始数据局部可积甚至无界Radon测度在$Q_{Ω,T},$中解的存在性.最后我们证明了除了热方程的局部正则化效应外，由于梯度项的存在，$u$还满足$L_{loc}^{R}% -L_{loc}^{\infty},$型的第二个效应.

英文摘要：

We obtain new a priori estimates for the nonnegative solutions of the equation \[ u_{t}-Δu+|\nabla u|^{q}=0 \] in $Q_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $Ω=\mathbb{R}^{N},$ or $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partialΩ\times\left( 0,T\right) .$ In case $Ω=\mathbb{R}^{N},$ we show that any solution $u\in C^{2,1}(Q_{\mathbb{R}^{N},T})$ of equation (1.1) in $Q_{\mathbb{R}^{N} ,T}$ (in particular any weak solution if $q\leqq2),$ without condition as $\left\vert x\right\vert \rightarrow\infty,$ satisfies the universal estimate \[ \left\vert \nabla u(.,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(.,t)}% {t},\qquad\text{in }Q_{\mathbb{R}^{N},T}. \] Moreover we prove that the growth of $u$ is limited by $C(t+t^{-1/(q-1}% )(1+\left\vert x\right\vert ^{q^{\prime}}),$ where $C$ depends on $u.$ We also give existence properties of solutions in $Q_{Ω,T},$ for initial data locally integrable or even unbounded Radon measures. We give a nonuniqueness result in case $q>2.$ Finally we show that besides the local regularizing effect of the heat equation, $u$ satisfies a second effect of type $L_{loc}^{R}% -L_{loc}^{\infty},$ due to the gradient term.

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