Trajectory attractors of energy balance climate models with bio-feedback (Q2908580)

The author considers the following functional reaction-diffusion problem with memory and a nonlocal Volterra operator, \(V=V(u)\). NEWLINE\[NEWLINEc(x) \frac{\partial u}{\partial t}-\nabla \cdot \left[k(x) |\nabla u|^{p-2} \nabla u\right] + g(u,V(u)) \in F(t,x,u,\overline{u},V(u)(t)), t>0, x \in M,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \overline{u}(t,x)=\int_{-T}^{0}\beta(s,x) u(t+s,x) ds,\;t>0,\;x \in MNEWLINE\]NEWLINE NEWLINE\[NEWLINE u(s,x)=u_{0}(s,x),\;-T \leq s \leq 0,\;x \in M,NEWLINE\]NEWLINE where \(M\) is a two-dimensional, compact, oriented Riemannian manifold without boundary of class \(C^{\infty}\), \(p>2\), \(c,k \in C^{2}(M)\) positive. This system is obtained from energy balance climate models describing the evolution of a ten-year mean temperature \(u\) through a balance of heat fluxes and modelling the ten-year mean of the horizontal heat flux as a diffusion operator. The reaction term \(F\) represents the absorbed radiation flux. The reaction term \(g\) represents the emitted radiation flux.NEWLINENEWLINEUnder certain assumptions on \(\beta, g, F, V\), the author proves the existence of a trajectory attractor for this system.