172 research outputs found
Self-similar solutions of the p-Laplace heat equation: the case when p>2
We study the self-similar solutions of the equation in when We make a complete study
of the existence and possible uniqueness of solutions of the form of any sign, regular
or singular at Among them we find solutions with an expanding compact
support or a shrinking hole (for or a spreading compact support or a
focussing hole (for When we show the existence of positive
solutions oscillating around the particular solution $U(x,t)=C_{N,p}(| x|
^{p}/(-t))^{1/(p-2)}.
A new dynamical approach of Emden-Fowler equations and systems
We give a new approach on general systems of the form (G){[c]{c}%
-\Delta_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=\epsilon_{1}|x|
^{a}u^{s}v^{\delta}, -\Delta_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla
u)=\epsilon_{2}|x|^{b}u^{\mu}v^{m}, where are
real parameters, and In
the radial case we reduce the problem to a quadratic system of order 4, of
Kolmogorov type. Then we obtain new local and global existence or nonexistence
results. In the case we also describe the
behaviour of the ground states in two cases where the system is variational. We
give an important result on existence of ground states for a nonvariational
system with and In the nonradial case we solve a conjecture of
nonexistence of ground states for the system with and and
Comment: 43 page
Stability properties for quasilinear parabolic equations with measure data
Let be a bounded domain of , and We study problems of the model type \left\{ \begin{array}
[c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on
}\partial\Omega\times(0,T),\\ u(0)=u_{0}\qquad\text{in }\Omega, \end{array}
\right. where , and Our main result is a \textit{stability theorem }extending the
results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for
quasilinear operators div\textit{. }Comment: arXiv admin note: substantial text overlap with arXiv:1310.525
Evolution equations of p-Laplace type with absorption or source terms and measure data
Let be a bounded domain of , and We consider problems\textit{ }of the type % \left\{
\begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u\pm\mathcal{G}(u)=\mu\qquad\text{in
}Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\
u(0)=u_{0}\qquad\text{in }\Omega, \end{array} \right. where
is the -Laplacian, is a bounded Radon measure, and is an absorption or a source term In
the model case
or has an exponential type. We prove the existence of
renormalized solutions for any measure in the subcritical case, and give
sufficient conditions for existence in the general case, when is good in
time and satisfies suitable capacitary conditions.Comment: arXiv admin note: substantial text overlap with arXiv:1310.525
Pointwise estimates and existence of solutions of porous medium and -Laplace evolution equations with absorption and measure data
Let be a bounded domain of . We obtain a
necessary and a sufficient condition, expressed in terms of capacities, for
existence of a solution to the porous medium equation with absorption
\begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{\Delta
}(|u|^{m-1}u)+|u|^{q-1}u=\mu ~ \text{in }\Omega \times (0,T), \\
{u}=0~~~\text{on }\partial \Omega \times (0,T), \\ u(0)=\sigma , \end{array}
\right. \end{equation*} where and are bounded Radon measures,
, . We also obtain a sufficient condition for
existence of a solution to the -Laplace evolution equation \begin{equation*}
\left\{ \begin{array}{l} {u_{t}}-{\Delta _{p}}u+|u|^{q-1}u=\mu ~~\text{in
}\Omega \times (0,T), \\ {u}=0 ~ \text{on }\partial \Omega \times (0,T), \\
u(0)=\sigma . \end{array} \right. \end{equation*} where and
An elliptic semilinear equation with source term and boundary measure data: the supercritical case
We give new criteria for the existence of weak solutions to an equation with
a super linear source term \begin{align*}-\Delta u = u^q
~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega\end{align*}where
is a either a bounded smooth domain or ,
q\textgreater{}1 and is a
nonnegative Radon measure on . One of the criteria we obtain is
expressed in terms of some Bessel capacities on . We also give
a sufficient condition for the existence of weak solutions to equation with
source mixed terms. \begin{align*} -\Delta u = |u|^{q\_1-1}u|\nabla u|^{q\_2}
~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega \end{align*} where
q\_1,q\_2\geq 0, q\_1+q\_2\textgreater{}1, q\_2\textless{}2, is a Radon measure on .Comment: Journal of Functional Analysis 269 (2015) 1995--201
On the connection between two quasilinear elliptic problems with source terms of order 0 or 1
We establish a precise connection between two elliptic quasilinear problems
with Dirichlet data in a bounded domain of The first one, of
the form
involves a source gradient term with natural growth, where is
nonnegative, , and is a nonnegative measure. The
second one, of the form
presents a source term of order where is nondecreasing, and is a
nonnegative measure. Here and can present an asymptote. The
correlation gives new results of existence, nonexistence, regularity and
multiplicity of the solutions for the two problems, without or with measures.
New informations on the extremal solutions are given when is superlinear
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