In the local case we will study the existence of wavefronts for the equations of the family

$$

u_t=Delta_p a(u) +
abla b(u) + c(u), p>1

$$

and later on these wavefronts will be used to determine the asymptotic behaviour of the solutions of the equation

$$

u_t = Delta_p u^m + h(u), m(p-1)>1, p>2,

$$

in dimension $Ngeq 1$, where $h(u)$ is a reaction term  (generally, either monostable, bistable or combustion type) whenever the solutions converge uniformly to 1, a question that will also be addressed in the session.

 

In the non-local case we will present two new models where two different operators are acting on different domains but are intertwined by a third non-local one, receiving the name of coupling models. The first model will present the coupling of a heat operator and an operator given by the convolution with a probability kernel, while the second will study the coupling of two fractional laplacians of different order coupled by a third one. In both cases questions about the existence of solutions, the decay rates of the $L^p$ norms of the solutions or the large time behaviour of them will be addressed, together with other questions about these novel models.