z-logo
Premium
The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations
Author(s) -
Han Wei
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3530
Subject(s) - mathematics , boundary value problem , neumann boundary condition , robin boundary condition , span (engineering) , mixed boundary condition , mathematical analysis , boundary (topology) , upper and lower bounds , civil engineering , engineering
In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u t t − u x x = b ( u , D u ) u x x + F ( u , D u ). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem [Disp. Item 1.1. 1.1 utt−uxx=b(u,Du)uxx+F(u,Du),x>0,t>0,t=0:u=εϕ(x),ut=εψ(x),x>0,x=0:∂u∂x=0,fort≥0, ...] is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span [Disp. Item 1.8. 1.8 T(ε)≥aε−α/2; ...] in the general case and [Disp. Item 1.10. 1.10 T(ε)≥aε−α(1+α)/(2+α);and ...] in the special case. Copyright © 2015 John Wiley & Sons, Ltd.

This content is not available in your region!

Continue researching here.

Having issues? You can contact us here