In this paper, we consider reaction-diffusion epidemic models with mass action or standard incidence mechanism and study the impact of limiting population movement on disease transmissions. We set either the dispersal rate of the susceptible or infected people to zero and study the corresponding degenerate reaction-diffusion model. Our main approach to study the global dynamics of these models is to construct delicate Lyapunov functions. Our results show that the consequences of limiting the movement of susceptible or infected people depend on transmission mechanisms, model parameters and population size.

New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features.

In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

We consider weak solutions $u:\Omega _T\to {\open R}^N$ to parabolic systems of the type

$$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$

$x\mapsto a(x,t,\xi )$

$D_xa(\cdot ,\cdot ,\xi )$

$L^\alpha (0,T;L^\beta (\Omega ))$

$\alpha ,\beta $

$$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$

$$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$

$u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$

Propagation at a finite speed is established for non-negative weak solutions to a thin-film approximation of the two-phase Muskat problem. The expansion rate of the support matches the scale invariance of the system. Moreover, we determine sufficient conditions on the initial data for the occurrence of waiting time phenomena.

If f0: Ω ⊂ ℝm → Sn is a weakly p-harmonic map from a bounded smooth domain Ω in ℝm (with 2 < p < m) into a sphere and if f0 is not stationary p-harmonic, then there exist infinitely many weak solutions of the p-harmonic flow with initial and boundary data f0, i.e., there are infinitely many global weak solutions f :Ω × ℝ → ⊂ Sn of

We also show that there exist non-stationary weakly (m − 1)-harmonic maps f0: Bm → Sm−1.