Gagliardo-Nirenberg type inequalities using fractional Sobolev spaces and Besov spaces

Our main purpose is to establish Gagliardo-Nirenberg type inequalities using fractional homogeneous Sobolev spaces, and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in [1, 2, 3, 7, 16, 21]

Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption

Here we study the initial trace problem for the nonnegative solutions of the equation $$u\_{t}-\Delta u+|\nabla u|^{q}=0$$ in $Q\_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $\Omega=\mathbb{R}^{N},$ or $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partial\Omega\times\left( 0,T\right) .$ We can define the trace at $t=0$ as a nonnegative Borel measure $(\mathcal{S} ,u\_{0}),$ where $S$ is the closed set where it is infinite, and $u\_{0}$ is a Radon measure on $\Omega\backslash\mathcal{S}.$ We show that the trace is a Radon measure when $q\leqq1.$ For $q\in(1,(N+2)/(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\_{0}.$ When $\mathcal{S}$ $=\overline{\omega}\cap\Omega$ and $\omega$ is an open subset of $\Omega,$ the existence extends to any $q\leqq2$ when $u\_{0}\in L\_{loc}^{1}(\Omega)$ and any $q>1$ when $u\_{0}=0$. In particular there exists a self-similar nonradial solution with trace $(\mathbb{R}^{N+},0),$ with a growth rate of order $\left\vert x\right\vert ^{q^{\prime}}$ as $\left\vert x\right\vert \rightarrow\infty$ for fixed $t.$ Moreover we show that the solutions with trace $(\overline{\omega},0)$ in $Q\_{\mathbb{R}^{N},T}$ may present near $t=0$ a growth rate of order $t^{-1/(q-1)}$ in $\omega$ and of order $t^{-(2-q)/(q-1)}$ on $\partial \omega.

On asymptotic properties of solutions to σ\sigma-evolution equations with general double damping

In this paper, we would like to consider the Cauchy problem for semi-linear $\sigma$-evolution equations with double structural damping for any $\sigma\ge 1$. The main purpose of the present work is to not only study the asymptotic profiles of solutions to the corresponding linear equations but also describe large-time behaviors of globally obtained solutions to the semi-linear equations. We want to emphasize that the new contribution is to find out the sharp interplay of ``parabolic like models" corresponding to $\sigma_1 \in [0,\sigma/2)$ and ``$\sigma$-evolution like models" corresponding to $\sigma_2 \in (\sigma/2,\sigma]$, which together appear in an equation. In this connection, we understand clearly how each damping term influences the asymptotic properties of solutions.Comment: 29 page

L∞L^{\infty} estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \left\{\begin{array} [c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. in $Q_{\Omega,T}=\Omega\times\left(0,T\right) ,$ where $q>1,\nu\geqq 0,T\in\left(0,\infty\right] ,$ and $\Omega=\mathbb{R}^{N}$ or $\Omega$ is a smooth bounded domain, and $u_{0}\in L^{r}(\Omega),r\geqq1,$ or $u_{0}\in\mathcal{M}_{b}(\Omega).$ We show $L^{\infty}$ decay estimates, valid for \textit{any weak solution}, \textit{without any conditions a}s $\left\| x\right\| \rightarrow\infty,$ and \textit{without uniqueness assumptions}. As a consequence we obtain new uniqueness results, when $u_{0}\in \mathcal{M}_{b}(\Omega)$ and $q<(N+2)/(N+1),$ or $u_{0}\in L^{r}(\Omega)$ and $q<(N+2r)/(N+r).$ We also extend some decay properties to quasilinear equations of the model type $$u_{t}-\Delta_{p}u+\left\| u\right\| ^{\lambda-1}u|\nabla u|^{q}=0$$ where $p>1,\lambda\geqq0,$ and $u$ is a signed solution

Isolated initial singularities for the viscous Hamilton-Jacobi equation

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi equation [u_{t}-\Delta u+|\nabla u|^{q}=0] in $Q_{\Omega,T}=\Omega\times(0,T),$ where $q>1,T\in(0,\infty] ,$ and $\Omega$ is a smooth bounded domain of $\mathbb{R}$ containing $0,$ or $\Omega=\mathbb{R}^{N}.$ We consider solutions with a possible singularity at point $(x,t)=(0,0).$ We show that if $q\geq q_{\ast}=(N+2)/(N+1)$ the singularity is removable.Comment: 32 page