arXiv subjects

Marie-Françoise Bidaut-Véron

Publications and source records attributed to Marie-Françoise Bidaut-Véron.

32 records · Page 2Linked to original sources

arXiv2014

Pointwise estimates and existence of solutions of porous medium and $p$-Laplace evolution equations with absorption and measure data

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}(N\geq 2)$. 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}}-{Δ}(|u|^{m-1}u)+|u|^{q-1}u=μ~ \text{in }Ω\times (0,T), \\ {u}=0~~~\text{on }\partial Ω\times (0,T), \\ u(0)=σ, \end{array} \right. \end{equation*} where $σ$ and $μ$ are bounded Radon measures, $q>\max (m,1)$, $m>\frac{N-2}{N}$. We also obtain a sufficient condition for existence of a solution to the $p$-Laplace evolution equation \begin{equation*} \left\{ \begin{array}{l} {u_{t}}-{Δ_{p}}u+|u|^{q-1}u=μ~~\text{in }Ω\times (0,T), \\ {u}=0 ~ \text{on }\partial Ω\times (0,T), \\ u(0)=σ. \end{array} \right. \end{equation*} where $q>p-1$ and $p>2$.

math.AP↗

arXiv2013

Stability properties for quasilinear parabolic equations with measure data and applications

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=Ω\times(0,T).$ We first study the problem \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where $p>1$, $μ\in\mathcal{M}_{b}(Ω)$ and $u_{0}\in L^{1}(Ω).$ Our main result is a \textit{stability theorem }extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case\textit{. } As an application, we consider the perturbed problem\textit{ } \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{Δ_{p}}u+\mathcal{G}(u)=μ\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partialΩ\times(0,T),\\ u(0)=u_{0}\qquad\text{in }Ω, \end{array} \right. \] where $\mathcal{G}(u)$ may be an absorption or a source term$.$ In the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$ or $\mathcal{G}$ has an exponential type. We give existence results when $q$ is subcritical, or when the measure $μ$ is good in time and satisfies suitable capacity conditions.

math.AP↗

arXiv2013

Keller-Osserman estimates for some quasilinear elliptic systems

In this article we study quasilinear multipower systems of two equations of two types, in a domain $Ω$ of R^{N} : with absorption terms, or mixed terms. Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type. Concerning the mixed system, we show that one of the solutions always satisfies Harnack inequality. In the case $Ω$=B(0,1)\{0}, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.

math.AP↗

arXiv2013

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

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-νΔu+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{Ω,T}=Ω\times\left(0,T\right) ,$ where $q>1,ν\geqq 0,T\in\left(0,\infty\right] ,$ and $Ω=\mathbb{R}^{N}$ or $Ω$ is a smooth bounded domain, and $u_{0}\in L^{r}(Ω),r\geqq1,$ or $u_{0}% \in\mathcal{M}_{b}(Ω).$ 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 $u_{0}\in \mathcal{M}_{b}(Ω)$ and $q<(N+2)/(N+1),$ or $u_{0}\in L^{r}(Ω)$ and $q<(N+2r)/(N+r).$ We also extend some decay properties to quasilinear equations of the model type \[ u_{t}-Δ_{p}u+\left\| u\right\| ^{λ-1}u|\nabla u|^{q}=0 \] where $p>1,λ\geqq0,$ and $u$ is a signed solution.

math.AP↗

arXiv2013

Remarks on some quasilinear equations with gradient terms and measure data

Let $Ω\subset \mathbb{R}^{N}$ be a smooth bounded domain, $H$ a Caratheodory function defined in $Ω\times \mathbb{R\times R}^{N},$ and $μ$ a bounded Radon measure in $Ω.$ We study the problem% \begin{equation*} -Δ_{p}u+H(x,u,\nabla u)=μ\quad \text{in}Ω,\qquad u=0\quad \text{on}\partial Ω, \end{equation*} where $Δ_{p}$ is the $p$-Laplacian ($p>1$)$,$ and we emphasize the case $H(x,u,\nabla u)=\pm \left\| \nabla u\right\| ^{q}$ ($q>0$). We obtain an existence result under subcritical growth assumptions on $H,$ we give necessary conditions of existence in terms of capacity properties, and we prove removability results of eventual singularities. In the supercritical case, when $μ\geqq 0$ and $H$ is an absorption term, i.e. $% H\geqq 0,$ we give two sufficient conditions for existence of a nonnegative solution.

math.AP↗

arXiv2013

Quasilinear Lane-Emden equations with absorption and measure data

We study the existence of solutions to the equation $-\Gd_pu+g(x,u)=μ$ when $g(x,.)$ is a nondecreasing function and $\gm$ a measure. We characterize the good measures, i.e. the ones for which the problem as a renormalized solution. We study particularly the cases where $g(x,u)=\abs x^β\abs u^{q-1}u$ and $g(x,u)=\abs x^τ\rm{sgn}(u)(e^{τ\abs u^λ}-1)$. The results state that a measure is good if it is absolutely continuous with respect to an appropriate Lorentz-Bessel capacities.

math.AP↗

arXiv2012

Isolated initial singularities for the viscous Hamilton-Jacobi equation

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi equation [u_{t}-Δu+|\nabla u|^{q}=0] in $Q_{Ω,T}=Ω\times(0,T),$ where $q>1,T\in(0,\infty] ,$ and $Ω$ is a smooth bounded domain of $\mathbb{R}% ^{N}$ containing $0,$ or $Ω=\mathbb{R}^{N}.$ 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.

math.AP↗

arXiv2011

Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion

In this article we study the positive solutions of the parabolic semilinear system of competitive type \[ \left\{\begin{array} [c]{c}% u_{t}-Δu+v^{p}=0, v_{t}-Δv+u^{q}=0, \end{array} \right. \] in $Ω\times\left(0,T\right) $, where $Ω$ is a domain of $\mathbb{R}^{N},$ and $p,q>0,$ $pq\neq1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form \[ u(x,t)\leqq Ct^{-(p+1)/(pq-1)},\qquad v(x,t)\leqq Ct^{-(q+1)/(pq-1)}% \] in $ω\times\left(0,T_{1}\right) ,$ for any domain $ω\subset\subsetΩ$ and $T_{1}\in\left(0,T\right) ,$ and $C=C(N,p,q,T_{1}% ,ω).$ For $p,q>1,$ we prove the existence of an initial trace at time 0, which is a Borel measure on $Ω.$ Finally we prove that the punctual singularities at time $0$ are removable when $p,q\geqq1+2/N.

math.AP↗

arXiv2010

Large solutions of elliptic systems of second order and applications to the biharmonic equation

In this work we study the nonnegative solutions of the elliptic system Δu=|x|^{a}v^δ, Δv=|x|^{b}u^μ in the superlinear case μδ>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point. In the radial case we give the precise behavior of the large solutions near the boundary in any dimension N. We also show the existence of infinitely many solutions blowing up at 0. Furthermore, we show that there exists a global positive solution in R^{N}\{0}, large at 0, and we describe its behavior. We apply the results to the sign changing solutions of the biharmonic equation Δ^2 u=|x|^{b}|u|^μ. Our results are based on a new dynamical approach of the radial system by means of a quadratic system of order 4, combined with nonradial upper estimates.

math.AP↗

arXiv2010

A new dynamical approach of Emden-Fowler equations and systems

We give a new approach on general systems of the form \[(G){[c]{c}% -Δ_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=ε_{1}|x| ^{a}u^{s}v^δ, -Δ_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=ε_{2}|x|^{b}u^μv^{m},\] where $Q,p,q,δ,μ,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $ε_{1}=\pm1,$ $ε_{2}=\pm1.$ 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 $ε_{1}=ε_{2}=1,$ 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 $p=q=2$ and $s=m>0.$ In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$ and $δ=m+1$ and $μ=s+1.$

math.AP↗

arXiv2009

Self-similar solutions of the p-Laplace heat equation: the case when p>2

We study the self-similar solutions of the equation \[ u_{t}-div(| \nabla u| ^{p-2}\nabla u)=0, \] in $\mathbb{R}^{N},$ when $p>2.$ We make a complete study of the existence and possible uniqueness of solutions of the form \[ u(x,t)=(\pm t)^{-α/β}w((\pm t)^{-1/β}| x|) \] of any sign, regular or singular at $x=0.$ Among them we find solutions with an expanding compact support or a shrinking hole (for $t>0),$ or a spreading compact support or a focussing hole (for $t<0).$ When $t<0,$ we show the existence of positive solutions oscillating around the particular solution $U(x,t)=C_{N,p}(| x| ^{p}/(-t))^{1/(p-2)}.$

math.AP↗

arXiv2008

Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of $\mathbb{R}^{N}.$ The first one, of the form $-Δ_{p}u=β(u)| \nabla u| ^{p}+λf(x),$ where $β$ is nonnegative, involves a gradient term with natural growth. The second one, of the form $-Δ_{p}v=λf(x)(1+g(v))^{p-1}$ where $g$ is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.

math.AP↗

arXiv2008

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 $\mathbb{R}^{N}.$ The first one, of the form \[ -Δ_{p}u=β(u)| \nabla u| ^{p}+λf(x)+α, \] involves a source gradient term with natural growth, where $β$ is nonnegative, $λ>0,f(x)\geqq0$, and $α$ is a nonnegative measure. The second one, of the form \[ -Δ_{p}v=λf(x)(1+g(v))^{p-1}+μ, \] presents a source term of order $0, $where $g$ is nondecreasing, and $μ$ is a nonnegative measure. Here $β$ and $g$ 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 $g$ is superlinear.

math.AP↗

arXiv2008

The p-Laplace heat equation with a source term : self-similar solutions revisited

We study the self-similar solutions of any sign of the equation u_{t}-div(|∇u|^{p-2}∇u)=|u|^{q-1}u, in R^{N}, where p,q>1. We extend the results of Haraux-Weissler obtained for p=2 to the case q>p-1>0. In particular we study the existence of slow or fast decaying solutions. For given t>0, the fast solutions u(t,.) have a compact support in R^{N} when p>2, and |x|^{p/(2-p)}u(t,x) is bounded at infinity when p<2. We describe the behaviour for large |x| of all the solutions. According to the position of q with respect to the first critical exponent p-1+p/N and the critical Sobolev exponent q^{∗}, we study the existence of positive solutions, or the number of the zeros of u(t,.). We prove that any solution u(t,.) is oscillatory when p<2 and q is closed to 1.

math.AP↗