scholarly journals Boundedness of global solutions of one dimensional quasilinear degenerate parabolic equations

1998 ◽  
Vol 50 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Ryuichi SUZUKI
Keyword(s):  
Parabolic Equations ◽  
Global Solutions ◽  
Degenerate Parabolic Equations ◽  
One Dimensional ◽  
Degenerate Parabolic

scholarly journals Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term

2019 ◽  
Vol 39 (3) ◽  
pp. 395-414
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Global Solutions ◽  
Positive Function ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Degenerate Parabolic Equations ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

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.

Complete quenching phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption

2019 ◽  
Vol 149 (5) ◽  
pp. 1323-1346 ◽  
Author(s):  
Nguyen Anh Dao ◽  
Jesus Ildefonso Díaz ◽  
Huynh Van Kha
Keyword(s):  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Degenerate Parabolic Equations ◽  
One Dimensional ◽  
Finite Speed Of Propagation ◽  
Nonnegative Solutions ◽  
Degenerate Parabolic ◽  
Speed Of Propagation ◽  
The One ◽  
Quenching Phenomenon

AbstractThis paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.

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