Uniqueness of Entropy Solutions of Nonlinear Elliptic-Parabolic-Hyperbolic Problems in One Dimension Space

We consider a class of elliptic-parabolic-hyperbolic degenerate equations of the form b(u)t — a(u, φ(ux)x= f with homogeneous Dirichlet conditions and initial conditions. In this paper we prove an L1-contraction principle and the uniqueness of entropy solutions under rather general assumptions on the data.We consider a class of elliptic-parabolic-hyperbolic degenerate equations of the form b(u)t — a(u, φ(ux)x= f with homogeneous Dirichlet conditions and initial conditions. In this paper we prove an L1-contraction principle and the uniqueness of entropy solutions under rather general assumptions on the data

Multivalued anisotropic problem with Fourier boundary condition involving diffuse Radon measure data and variable exponents

We study a nonlinear anisotropic elliptic problem under Fourier type boundary condition governed by a general anisotropic operator with variable exponents and diffuse Radon measure data which does not charge the sets of zero p(·)-capacity. We prove an existence and uniqueness result of entropy or renormalized solution

Suitable Radon measure for nonlinear Dirichlet boundary p(u)-Laplacian problem

This paper is devoted to the study of nonlinear homogeneous Dirichlet boundary p(u)-laplacian problem. Existence, uniqueness and structural stability results of weak solutions are obtained by approximation method and convergent sequences in terms of Young measures

On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination

We study a vector-borne disease with age of vaccination. A nonlinear incidence rate including mass action and saturating incidence as special cases is considered. The global dynamics of the equilibria are investigated and we show that if the basic reproduction number is less than 1, then the disease-free equilibrium is globally asymptotically stable; that is, the disease dies out, while if the basic reproduction number is larger than 1, then the endemic equilibrium is globally asymptotically stable, which means that the disease persists in the population. Using the basic reproduction number, we derive a vaccination coverage rate that is required for disease control and elimination

Structural stability for variable exponent elliptic problems. I. The p(x)p(x)-laplacian kind problems.

the first version of the preprint is now split into two partsInternational audienceWe study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b(u_n)-\div mathfrak{a}_n(x,\Grad u_n)=f_n. The equation is set in a bounded domain \Om of $\R^N$ and supplied with the homogeneous Dirichlet boundary condition on \ptl\Om. Here $b$ is a non-decreasing function on $\R$, and $\Bigl(\mathfrak{a}_n(x,\xi)\Bigr)_n$ is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent $p_n(x)$, $

Well-posedness results for triply nonlinear degenerate parabolic equations

We study the well-posedness of triply nonlinear degenerate elliptic-parabolic-hyperbolic problem $$b(u)_t - {\rm div} \tilde{\mathfrak a}(u,\nabla\phi(u))+\psi(u)=f, \quad u|_{t=0}=u_0$$ in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities $b,\phi$ and $\psi$ are supposed to be continuous non-decreasing, and the nonlinearity $\tilde{\mathfrak a}$ falls within the Leray-Lions framework. Some restrictions are imposed on the dependence of $\tilde{\mathfrak a}(u,\nabla\phi(u))$ on $u$ and also on the set where $\phi$ degenerates. A model case is $\tilde{\mathfrak a}(u,\nabla\phi(u)) =\tilde{\mathfrak{f}}(b(u),\psi(u),\phi(u))+k(u)\mathfrak{a}_0(\nabla\phi(u)),$ with $\phi$ which is strictly increasing except on a locally finite number of segments, and $\mathfrak{a}_0$ which is of the Leray-Lions kind. We are interested in existence, uniqueness and stability of entropy solutions. If $b=\mathrm{Id}$, we obtain a general continuous dependence result on data $u_0,f$ and nonlinearities $b,\psi,\phi,\tilde{\mathfrak{a}}$. Similar result is shown for the degenerate elliptic problem which corresponds to the case of $b\equiv 0$ and general non-decreasing surjective $\psi$. Existence, uniqueness and continuous dependence on data $u_0,f$ are shown when $[b+\psi](\R)=\R$ and $\phi\circ [b+\psi]^{-1}$ is continuous

Structural stability for variable exponent elliptic problems. II. The p(u)p(u)-laplacian and coupled problems.

International audienceWe study well-posedness for elliptic problems under the form b(u)-\div \mathfrak{a}(x,u,\Grad u)=f, where $\mathfrak{a}$ satisfies the classical Leray-Lions assumptionswith an exponent $p$ that may depend both on the space variable $x$ and on the unknown solution $u$. A prototype case is the equation u-\div \Bigl( |\grad u|^{p(u)-2}\grad u \Bigr)=f. We have to assume that \inf_{x\in\overline{\Om},\,z\in\R} p(x,z) is greater than the space dimension $N$. Then, under mild regularity assumptions on \Om and on the nonlinearities, we show that the associated solution operator is an order-preserving contraction in L^1(\Om). In addition, existence analysis for a sample coupled system for unknowns $(u,v)$ involving the $p(v)$-laplacian of $u$ is carried out. Coupled elliptic systems with similar structure appear in applications, e.g. in modelling of stationary thermo-rheological fluids

Numerical analysis of nonlinear parabolic problems with variable exponent and L1L^1 data

In this paper, we make the numerical analysis of the mild solution which is also an entropy solution of parabolic problem involving the $p(x)-$Laplacian operator with $L^1-$ data