Asymptotic profiles of solutions to convection–diffusion equations

The large time behavior of zero-mass solutions to the Cauchy problem for the convection–diffusion equation $\text{u}{}_{\mathrm{t}}\text{?u}{}_{\mathrm{xx}}\text{+(|u|}{}^{\mathrm{q}}\text{)}{}_{\mathrm{x}}\text{=0,}\text{u(x,0)=u}{}_0\text{(x)}$ is studied when q>1 and the initial datum u₀ belongs to $\text{L}{}^1\text{(}\text{R}\text{,(1+|x|)}\text{d}\text{x)}$ and satisfies $\text{∫}{}_{\mathrm{<mtext>R</mtext>}}\text{u}{}_0\text{(x)}\text{d}\text{x=0}$. We provide conditions on the size and shape of the initial datum u₀ as well as on the exponent q>1 such that the large time asymptotics of solutions is given either by the derivative of the Gauss–Weierstrass kernel, or by a self-similar solution of the equation, or by hyperbolic N-waves. To cite this article: S. Benachour et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Global existence and blowing-up of solutions to a class of coupled reaction–convection–diffusion systems

This paper concerns the global existence and blowing-up of solutions to the homogeneous Neumann problem of a coupled reaction–convection–diffusion system. The critical Fujita curve is determined and blowing-up theorem of Fujita type is established. An interesting phenomenon is that the critical Fujita curve even could be the infinite due to the convection.     

Numerical solutions for convection–diffusion equation using El-Gendi method

The method of El-Gendi [El-Gendi SE. Chebyshev solution of differential integral and integro-differential equations. J Comput 1969;12:282–7; Mihaila B, Mihaila I. Numerical approximation using Chebyshev polynomial expansions: El-gendi’s method revisited. J Phys A Math Gen 2002;35:731–46] is presented with interface points to deal with linear and non-linear convection–diffusion equations.The linear problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using three-level time scheme.The non-linear problem is reduced to three systems of ordinary differential. Each one of these systems is, then, solved using three-level time scheme. Numerical results for Burgers’ equation and modified Burgers’ equation are shown and compared with other methods. The numerical results are found to be in good agreement with the exact solutions.     

Special features of strongly coupled systems of convection–diffusion equations with two small parameters

Strong coupling of convection–diffusion equations with two small parameters generates a solution decomposition which differs significantly from that for the one-parameter case. We explain the basic features and prove pointwise estimates for the first-order derivatives which allow us to analyze the upwind finite difference scheme on layer-adapted meshes.     

Some classes of inverse problems of determining the source function in convection–diffusion systems

We consider the generalized solvability of convection–diffusion systems and study the inverse problem of determining the right-hand side (the source function) of such a system from integral overdetermination data. The solution of a parabolic system is understood in the generalized sense, and distributions of certain classes are allowed as right-hand sides. Under certain conditions on the problem data, we show that the inverse problem is well-posed in the Sobolev classes.     

Curves of steepest descent are entropy solutions for a class of degenerate convection–diffusion equations

We consider a nonlinear degenerate convection–diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kru?kov are obtained as the—a posteriori unique—limit points of the JKO variational approximation scheme for an associated gradient flow in the $L^2$ -Wasserstein space. The equation lacks the necessary convexity properties which would allow to deduce well-posedness of the initial value problem by the abstract theory of metric gradient flows. Instead, we prove the entropy inequality directly by variational methods and conclude uniqueness by doubling of the variables.     

Stability of solutions of control problems for the convection–diffusion–reaction equation with a strong nonlinearity

We consider a boundary control problem for the stationary convection–diffusion–reaction equation in which the reaction constant depends on the concentration of matter in such a way that the equation has a fifth-order nonlinearity. We prove the solvability of the boundary value problem and an extremal problem, derive an optimality system, and analyze it to derive estimates for the local stability of the solution of the extremal problem under small perturbations of both the performance functional and one of the given functions.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

Blowup and self-similar solutions for two-component drift–diffusion systems

We discuss asymptotic properties of solutions of two-component parabolic drift–diffusion systems coupled through an elliptic equation in two space dimensions. In particular, conditions for finite time blowup versus the existence of forward self-similar solutions are studied.     

Instability induced by cross-diffusion in reaction–diffusion systems

In this paper the instability of the uniform equilibrium of a general strongly coupled reaction–diffusion is discussed. In unbounded domain and bounded domain the sufficient conditions for the instability are obtained respectively. The conclusion is applied to the ecosystem, it is shown that cross-diffusion can induce the instability of an equilibrium which is stable for the kinetic system and for the self-diffusion–reaction system.     

Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection–reaction–diffusion systems. Under certain conditions, we prove that there exists a maximal wave speed c^?$c^{?}$ such that for each wave speed c≤c^?$c\le c^{?}$, there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c≤c^?$c\le c^{?}$ are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves with speed c>c^?$c>c^{?}$.