American Institute of Mathematical Sciences

$\varepsilon u_\varepsilon''+ u_\varepsilon'+m(|A^{1/2}u_\varepsilon|^2)Au_\varepsilon=0, \quad u_\varepsilon(0)=u_0,\quad u_\varepsilon'(0)=u_1,$

and the first order limit problem

$u'+m(|A^{1/2}u_\varepsilon|^2)Au=0, \quad u(0)=u_0,$

where $\varepsilon>0$, $H$ is a Hilbert space, $A$ is a self-adjoint nonnegative operator on $H$ with dense domain $D(A)$, $(u_0,u_1)\in D(A^{3/2})\times D(A^{1/2})$, and $m:[0,+\infty)\to [0,+\infty)$ is a function of class $C^1$.
We prove global-in-time estimates for the difference $u_\varepsilon(t)-u(t)$ provided that $u_0$ satisfies the nondegeneracy condition $m(|A^{1/2}u_0|^2)>0$, and the function $\sigma m(\sigma^2)$ is nondecreasing in a right neighborhood of its zeroes.
The abstract results apply to parabolic and hyperbolic partial differential equations with non-local nonlinearities of Kirchhoff type.

$\partial_t u = \Delta u +f(u)$ in $R^N \times (0,\infty),$

$u (x,0) = \phi (x) \ge 0$ in $R^N,\quad\quad$

where $N \geq 1$, $f \in C^1([0,\infty))$, and $\phi \in L^1(R^N) \cap L^{\infty}(R^N)$. We study the asymptotic behavior of the solutions in the $L^q$ spaces with $q \in [1,\infty]$, by using the relative entropy methods.