827 research outputs found
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 in
where
and or is a smooth bounded domain of
and on We can
define the trace at as a nonnegative Borel measure where is the closed set where it is infinite, and is a
Radon measure on We show that the trace is a
Radon measure when For and any given Borel
measure, we show the existence of a minimal solution, and a maximal one on
conditions on When and
is an open subset of the existence extends to any
when and any when . In
particular there exists a self-similar nonradial solution with trace
with a growth rate of order as for fixed
Moreover we show that the solutions with trace in
may present near a growth rate of order
in and of order on $\partial
\omega.
On asymptotic properties of solutions to -evolution equations with general double damping
In this paper, we would like to consider the Cauchy problem for semi-linear
-evolution equations with double structural damping for any . 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 and ``-evolution like models" corresponding to , 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
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
where and or is a
smooth bounded domain, and or We show decay estimates, valid for
\textit{any weak solution}, \textit{without any conditions a}s and \textit{without uniqueness assumptions}. As a
consequence we obtain new uniqueness results, when and or and
We also extend some decay properties to quasilinear equations
of the model type where and 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
where and is a smooth bounded domain of
containing or We consider
solutions with a possible singularity at point We show that if
the singularity is removable.Comment: 32 page
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