Null Condition for Semilinear Wave Equation with Variable Coefficients

Catalano, Fabio. "Null Condition for Semilinear Wave Equation with Variable Coefficients." Serdica Mathematical Journal 25.4 (1999): 321-340. <http://eudml.org/doc/11522>.

@article{Catalano1999,
abstract = {∗The author was partially supported by M.U.R.S.T. Progr. Nazionale “Problemi Non Lineari...”In this work we analyse the nonlinear Cauchy problem (∂tt − ∆)u(t, x) = ( λg + O(1/(1 + t + |x|)^a) ) ) ∇t,x u(t, x), ∇t,x u(t, x) ), whit initial data u(0, x) = e u0 (x), ut (0, x) = e u1 (x). We assume a ≥ 1, x ∈ R^n (n ≥ 3) and g the matrix related to the Minkowski space. It can be considerated a pertubation of the case when the quadratic term has constant coefficient λg (see Klainerman [6]) We prove a global existence and uniqueness theorem for very regular initial data. The proof avoids a direct application of Klainermann method (Null condition, energy conformal method), because the result is obtained by a combination beetwen the energy estimate (norm L^2 ) and the decay estimate (norm L^∞ ).},
author = {Catalano, Fabio},
journal = {Serdica Mathematical Journal},
keywords = {compactly supported initial data; decay estimate},
language = {eng},
number = {4},
pages = {321-340},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Null Condition for Semilinear Wave Equation with Variable Coefficients},
url = {http://eudml.org/doc/11522},
volume = {25},
year = {1999},
}

TY - JOUR
AU - Catalano, Fabio
TI - Null Condition for Semilinear Wave Equation with Variable Coefficients
JO - Serdica Mathematical Journal
PY - 1999
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 25
IS - 4
SP - 321
EP - 340
AB - ∗The author was partially supported by M.U.R.S.T. Progr. Nazionale “Problemi Non Lineari...”In this work we analyse the nonlinear Cauchy problem (∂tt − ∆)u(t, x) = ( λg + O(1/(1 + t + |x|)^a) ) ) ∇t,x u(t, x), ∇t,x u(t, x) ), whit initial data u(0, x) = e u0 (x), ut (0, x) = e u1 (x). We assume a ≥ 1, x ∈ R^n (n ≥ 3) and g the matrix related to the Minkowski space. It can be considerated a pertubation of the case when the quadratic term has constant coefficient λg (see Klainerman [6]) We prove a global existence and uniqueness theorem for very regular initial data. The proof avoids a direct application of Klainermann method (Null condition, energy conformal method), because the result is obtained by a combination beetwen the energy estimate (norm L^2 ) and the decay estimate (norm L^∞ ).
LA - eng
KW - compactly supported initial data; decay estimate
UR - http://eudml.org/doc/11522
ER -