TY  - JOUR
ID  - Nakao2019
LB  - Nakao2019
AU  - Nakao, Mitsuhiro
TI  - Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term
ST  - Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2019
VL  - 39
IS  - 3
SP  - 395
EP  - 414
DO  - https://doi.org/10.7494/OpMath.2019.39.3.395
UR  - https://doi.org/10.7494/OpMath.2019.39.3.395
AB  - We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given.
KW  - degenerate quasilinear parabolic equation
KW  - nonlinear source term
KW  - Moser's method
SN  - 1232-9274
ER  -