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Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials
Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates
| 1. | Dipartimento di Matematica, University of Pisa, Italy |
| 2. | Dipartimento di Matematica Applicata, University of Pisa, Italy |
$\varepsilon u_\varepsilon''+ u_\varepsilon'+m(|A^{1/2}u_\varepsilon|^2)Au_\varepsilon=0, \quad u_\varepsilon(0)=u_0,\quad u_\varepsilon'(0)=u_1,$
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.
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