Renormalised solutions to reaction-diffusion systems with interface conditions: Global existence and weak-strong uniqueness
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen（格拉茨大学数学与科学计算研究所）
We introduce an extension of the concept of renormalised solutions for entropy-dissipating reaction-diffusion systems due to J. Fischer (Arch. Ration. Mech. Anal. 218, 2015) to systems coupled by nonlinear interface conditions. For this notion of solution, we establish global existence as well as a weak-strong stability estimate. Our framework allows to handle entropy-dissipating interfacial transmission rates without growth restrictions, including power-law nonlinearities as arising in the thermodynamic modelling of dissipative bulk-interface systems via generalised gradient structures. Our analysis relies on suitable extensions of the species' densities across the interface as well as on a non-local truncated variant of the relative entropy.
The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with no…
We establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. The weak-strong uniqueness principle holds for dissipative renormalised solutions, which in addition to the renormalised formulation obey suitable dissipation inequalities consistent with previous existence results. We treat general entropy-dissipating reactions without growt…
The convergence to equilibrium for renormalised solutions to nonlinear reaction-diffusion systems is studied. The considered reaction-diffusion systems arise from chemical reaction networks with mass action kinetics and satisfy the complex balanced condition. By applying the so-called entropy method, we show that if the system does not have boundary equilibria, then any renormalised solution converges exponentially to the complex balanced equilibrium with a rate, which can be computed explicitly up to a finite dimensional inequality. This inequality is proven via a contradiction argument and t…
In this paper our goal is to define a renormalised dissipative measure--valued (rDMV) solution of compressible Navier--Stokes system for fluids with non-monotone pressure--density relation. We prove existence of rDMV solutions and establish a suitable relative energy inequality. Moreover we obtain the Weak (Measure-valued)-Strong uniqueness property of this rDMV solution with the help of relative energy inequality.
In this paper, we consider the inverse reaction coefficient problems (IRCPs) for nonlocal diffusion equation and multi-term time-fractional nonlocal diffusion equation from the average nonlocal flux data in external reaction region. Based on the nonlocal maximum principle we established, the uniqueness theorem for IRCPs are proved.
We establish the uniqueness of semi-wavefront solution for a non-local delayed reaction-diffusion equation. This result is obtained by using a generalization of the Diekman-Kaper theory for a nonlinear convolution equation. Several applications to the systems of non-local reaction-diffusion equations with distributed time delay are also considered.
We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper $L^{\infty}$-bounds that exploits the entropic gradient structure of the system.
We prove pathwise uniqueness for an abstract stochastic reaction-diffusion equation in Banach spaces. The drift contains a bounded Hölder term; in spite of this, due to the space-time white noise it is possible to prove pathwise uniqueness. The proof is based on a detailed analysis of the associated Kolmogorov equation. The model includes examples not covered by the previous works based on Hilbert spaces or concrete SPDEs.
The paper deals with reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. Such problems describe biological processes and chemical reactions in which diffusive and nondiffusive substances interact according to hysteresis law. Under the assumption that the initial data are spatially transverse, we prove a theorem on the uniqueness of solutions. The theorem covers the case of non-Lipschitz hysteresis branches arising in the theory of slow-fast systems.
In this paper, we provide a mathematical framework in studying the wave propagation with the annihilation phenomenon in excitable media. We deal with the existence and uniqueness of solutions to a one-dimensional free boundary problem (called a reaction--interface system) arising from the singular limit of a FitzHugh--Nagumo type reaction--diffusion system. Because of the presence of the annihilation, interfaces may intersect each other. We introduce the notion of weak solutions to study the continuation of solutions beyond the annihilation time. Under suitable conditions, we show that the fre…
We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic reactions the fronts are proved to propagate with a deterministic positive speed. Our results extend to reaction-advection-diffusion equations with periodic advection and diffusion.
We establish a weak-strong uniqueness principle for solutions to entropy-dissipating reaction-diffusion equations: As long as a strong solution to the reaction-diffusion equation exists, any weak solution and even any renormalized solution must coincide with this strong solution. Our assumptions on the reaction rates are just the entropy condition and local Lipschitz continuity; in particular, we do not impose any growth restrictions on the reaction rates. Therefore, our result applies to any single reversible reaction with mass-action kinetics as well as to systems of reversible reactions wit…
The weak-strong uniqueness for solutions to reaction-cross-diffusion systems in a bounded domain with no-flux boundary conditions is proved. The system generalizes the Shigesada-Kawasaki-Teramoto population model to an arbitrary number of species. The diffusion matrix is neither symmetric nor positive definite, but the system possesses a formal gradient-flow or entropy structure. No growth conditions on the source terms are imposed. It is shown that any renormalized solution coincides with a strong solution with the same initial data, as long as the strong solution exists. The proof is based o…
In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $μ_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements o…
We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with co…
We study the question of uniqueness of weak solution to the fast reaction limit of the Keller and Rubinow model for Liesegang rings as introduced by Hilhorst et al. (J. Stat. Phys. 135, 2009, pp. 107-132). The model is characterized by a discontinuous reaction term which can be seen as an instance of spatially distributed non-ideal relay hysteresis. In general, uniqueness of solutions for such models is conditional on certain transversality conditions. For the model studied here, we give an explicit description of the precipitation boundary which gives rise to two scenarios for non-uniqueness,…
We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.
In this work, we study the mathematical analysis of a coupled system of two reaction-diffusion-advection equations and Danckwerts boundary conditions, which models the interaction between a microbial population (e.g., bacterias) and a diluted substrate (e.g., nitrate) in a continuous flow bioreactor. This type of bioreactor can be used, for instance, for water treatment. First, we prove the existence and uniqueness of solution, under the hypothesis of linear reaction by using classical results for linear parabolic boundary value problems. Next, we prove the existence and uniqueness of solution…
We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profil…
In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + σ(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $σ$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence …
This paper concerns with existence, uniqueness and asymptotic behavior of the solutions for a nonlocal coupled system of reaction-diffusion. We prove the existence and uniqueness of weak solutions by the Faedo-Galerkin method and exponential decay of solutions by the classic energy method. We improve the results obtained by Chipot-Lovato and Menezes for coupled systems. A numerical scheme is presented.
We consider positive travelling fronts of the time-delayed reaction-diffusion equation with the monostable birth function. Our main result says that for every fixed and sufficiently large velocity c, the positive travelling front is unique (modulo translations). To prove the uniqueness, we introduce a small parameter 1/c and realize the Lyapunov-Schmidt reduction in a scale of Banach spaces.
This paper is concerned with an age-structured model in population dynamics. We investigate the uniqueness of solution for this type of nonlinear reaction-diffusion problem when the source term depends on the density, indicating the presence of, for example, mortality and reaction processes. Our result shows that in a spatial environment, if two population densities obey the same evolution equation and possess the same terminal data of time and age, then their distributions must coincide therein.
Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$ converges to $-\infty$ at a super-linear rate as $u\to\infty$. We give a sharp minimal growth condition on $f$, independent of $L$, in order that there exist a universal, a priori upper bound for all solutions to the above Cauch…
We consider the Cauchy problems of a non-strictly hyperbolic system which describes the compressible Euler fluid with exothermic reaction. In this paper a Lyapunov-type functional is constructed for balance laws. By analysis of the flow generated by front tracking method, we prove the well-posedness theorems and present the local features of global solutions.
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn-Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in $u$ reaction term $g(x,t,u).$
We study existence and uniqueness of travelling fronts, and asymptotic speed of propagation for a non local reaction diffusion equation with spatial and genetic trait structure.
In this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient. Under quite general assumptions on the convection term we prove the existence of a weak solution by applying the theory of pseudomonotone operators. Imposing some linear conditions on the gradient variable the uniqueness of the solution is obtained.
We consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [11]. Here we prove the uniqueness of weak solutions.
We prove the existence and uniqueness of a traveling front and of its speed for the homogeneous heat equation in the half-plane with a Neumann boundary reaction term of non-balanced bistable type or of combustion type. We also establish the monotonicity of the front and, in the bistable case, its behavior at infinity. In contrast with the classical bistable interior reaction model, its behavior at the side of the invading state is of power type, while at the side of the invaded state its decay is exponential. These decay results rely on the construction of a family of explicit bistable traveli…
In this paper we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising in the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the end time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an a…
We study the uniqueness of steady states of strong-KPP reaction--diffusion equations in general domains under various boundary conditions. We show that positive bounded steady states are unique provided the domain satisfies a certain spectral nondegeneracy condition. We also formulate a number of open problems and conjectures.
We consider the semi linear Fisher-Kolmogorov-Petrovski-Piscounov equation for the advance of an advantageous gene in biology. Its non-smooth reaction function $f(u)$ allows for the introduction of travelling waves with a new profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions $u(x,t)$, $(x,t)\in \mathbb{R}\times \mathbb{R}_+$. We prove also the existence and uniqueness (up to a spatial shift) of a travelling wave $U$. Our main result is the uniform convergence (for $x\in \mathbb{R}$) of every solution $u(x,t)$ of the Cauchy problem to a single traveling…
We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a continuation method which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.
The paper deals with reaction-diffusion equations involving a hysteretic discontinuity in the source term, which is defined at each spatial point. In particular, such problems describe chemical reactions and biological processes in which diffusive and nondiffusive substances interact according to hysteresis law. We find sufficient conditions that guarantee the existence and uniqueness of solutions as well as their continuous dependence on initial data.
We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.
We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected 1-dimensional media with Dirichlet conditions outside the domain. After demonstrating the existence and uniqueness of a non-trivial bounded stationary state $n\_+$ , we prove that the stable state $n\_+$ invades the unstable state 0 with a speed which is exponential in time.
We show global well-posedness and exponential stability of equilibria for a general class of nonlinear dissipative bulk-interface systems. They correspond to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes nonlinear slow and fast diffusion in the bulk and nonlinear coupled diffusion on the interface. Additional driving mechanisms can be included and non-smooth geometries and coefficients are admissible, to some extent. An important application are volume-surface reaction-diffusion systems with nonlinear coupled diffusion.
In this short paper I report on a paper published in Nonlinear Analysis: Real World Applications in 2004. There is a major mistake early in that paper which makes most of its claims false. The class of reaction-diffusion systems considered in the paper has been the object of a renewed investigation in the past few years, by myself and others, and recent discoveries provide explicit counterexamples .
We prove a contraction in $L^1$ property for the solutions of a nonlinear reaction--diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as $t$ tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution.
In this work, we consider a reaction-diffusion system, modeling the interaction between nutrients, phytoplanktons and zooplanktons. Using a semigroup approach in $L^2$, we prove global existence, uniqueness and positivity of the solutions. The Holling type 2 nonlinearities, i.e of rational type with singularity, are handled by providing estimates in $L^\infty$. The article finally exhibits some time asymptotics properties of the solutions.
We prove existence and uniqueness of \emph{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in $\real^N$, with $m>1$, $0<p<1$ and the critical value for the weight $$ σ=\frac{2(1-p)}{m-1}. $$ Existence and uniqueness of some specific solution holds true when $m+p\geq2$. On the contrary, no eternal solution exists if $m+p<2$. We also classify exponential self-similar solutions with a different interface behavior when $m+p>2$. Some transformations to reaction-convection-diffusion e…
This work focuses on dynamics arising from reaction-diffusion equations , where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment, under very wide and generic conditions. We are especially concerned with their existence, unique…
This paper is devoted to reaction-diffusion equations with bistable nonlinearities depending periodically on time. These equations admit two linearly stable states. However, the reaction terms may not be bistable at every time. These may well be a periodic combination of standard bistable and monostable nonlinearities. We are interested in a particular class of solutions, namely pulsating fronts. We prove the existence of such solutions in the case of small time periods of the nonlinearity and in the case of small perturbations of a nonlinearity for which we know there exist pulsating fronts. …
This paper explains the uniqueness of positive steady state of general Lotka-Volterra competition model of two species of animals in the same environment.
We study traveling waves for a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions. We describe relations between speeds and asymptotic of profiles of traveling waves, and prove the uniqueness of the profiles up to shifts.
We study reaction-diffusion equations in one spatial dimension and with general (space- or time-) inhomogeneous mixed bistable-ignition reactions. For those satisfying a simple quantitative hypothesis, we prove existence and uniqueness of transition fronts, as well as convergence of "typical" solutions to the unique transition front (the existence part even extends to mixed bistable-ignition-monostable reactions). These results also hold for all pure ignition reactions without any other hypotheses, but not for all pure bistable reactions. In fact, we find examples of either spatially or tempor…
We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = Δu(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a simi…
