We study the following one-dimensional evolution equation: \[\frac{\partial u}{\partial t}(x,t)=\int_{A^{^{+}}u(x,t)}\lambda_{1}(\xi,t)\left(u(\xi,t)-u(x,t)\right)d\xi-\int_{A^{^{-}}u(x,t)}\lambda_{2}(\xi,t)\left(u(x,t)-u(\xi,t)\right)d\xi\] where $A^{^{+}}u(x,t)=\{\xi\in[0,1]\,|\, u(\xi,t)>u(x,t)\},\,\, A^{^{-}}u(x,t)=[0,1]\backslash A^{^{+}}u(x,t)$, and $\lambda_{1}$, $\lambda_{2}$ are non-negative functions. We prove existence of solutions for a particular class of initial data $u(x,0).$ We also prove that solutions are unique. Finally, under additional constraints on the initial data, we give an explicit expression for the solution.
Existence and unicity of solutions for a non-local relaxation equation
PAPARELLA, Francesco;PASCALI, Eduardo
2009-01-01
Abstract
We study the following one-dimensional evolution equation: \[\frac{\partial u}{\partial t}(x,t)=\int_{A^{^{+}}u(x,t)}\lambda_{1}(\xi,t)\left(u(\xi,t)-u(x,t)\right)d\xi-\int_{A^{^{-}}u(x,t)}\lambda_{2}(\xi,t)\left(u(x,t)-u(\xi,t)\right)d\xi\] where $A^{^{+}}u(x,t)=\{\xi\in[0,1]\,|\, u(\xi,t)>u(x,t)\},\,\, A^{^{-}}u(x,t)=[0,1]\backslash A^{^{+}}u(x,t)$, and $\lambda_{1}$, $\lambda_{2}$ are non-negative functions. We prove existence of solutions for a particular class of initial data $u(x,0).$ We also prove that solutions are unique. Finally, under additional constraints on the initial data, we give an explicit expression for the solution.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
