CPAA
Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms
Olivier Guibé Anna Mercaldo
Communications on Pure & Applied Analysis 2008, 7(1): 163-192 doi: 10.3934/cpaa.2008.7.163
In the present paper we prove uniqueness results for weak solutions to a class of problems whose prototype is

$-d i v((1+|\nabla u|^2)^{(p-2)/2} \nabla u)-d i v(c(x) (1+|u|^2)^{(\tau+1)/2}) $

$+b(x) (1+|\nabla u|^2)^{(\sigma+1)/2}=f \ i n \ \mathcal D'(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

$u\in W^{1,p}_0(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

where $\Omega$ is a bounded open subset of $\mathbb R^N$ $(N\ge 2)$, $p$ is a real number $\frac{2N}{N+1}< p <+\infty$, the coefficients $c(x)$ and $b(x)$ belong to suitable Lebesgue spaces, $f$ is an element of the dual space $W^{-1,p'}(\Omega)$ and $\tau$ and $\sigma$ are positive constants which belong to suitable intervals specified in Theorem 2.1, Theorem 2.2 and Theorem 2.3.

keywords: nonlinear elliptic equations Uniqueness noncoercive problems weak solutions.
CPAA
Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation
Dominique Blanchard Nicolas Bruyère Olivier Guibé
Communications on Pure & Applied Analysis 2013, 12(5): 2213-2227 doi: 10.3934/cpaa.2013.12.2213
We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq's systems. The main tools rely on regularity results for the Navier-Stokes equations with precise estimates on the solution with respect to the data in dimension $2$ and on the techniques involved for renormalized solutions of parabolic problems.
keywords: Navier-Stokes equations parabolic equations. Nonlinear systems of PDE
CPAA
Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities
Dominique Blanchard Olivier Guibé Hicham Redwane
Communications on Pure & Applied Analysis 2016, 15(1): 197-217 doi: 10.3934/cpaa.2016.15.197
In this paper we prove the existence and uniqueness of a renormalized solution for nonlinear parabolic equations whose model is \begin{eqnarray} \frac{\partial b(u)}{\partial t} - div\big(a(x,t,u,\nabla u)\big)=f+ div (g), \end{eqnarray} where the right side belongs to $L^{1}(Q)+L^{p'}(0,T;W^{-1,p'}(\Omega))$, where $b(u)$ is a real function of $u$ and where $-div(a(x,t,u,\nabla u))$ is a Leray-Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$, but without any growth assumption on $u$.
keywords: Nonlinear parabolic equations uniqueness renormalized solutions. existence

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