scholarly journals Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity

2014 ◽  
Vol 34 (3) ◽  
pp. 569 ◽  
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Viscoelastic Equation ◽  
Initial Boundary Value

Energy Decay of Global Solutions for a System of Petrovsky Equations

2013 ◽  
Vol 405-408 ◽  
pp. 3160-3164
Author(s):  
Yao Jun Ye
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Decay Estimate ◽  
Boundary Value ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Energy Decay ◽  
Difference Inequality ◽  
Initial Boundary Value

The initial-boundary value problem for a class of nonlinear Petrovsky systems in bounded domain is studied. We prove the energy decay estimate of global solutions through the use of a difference inequality.

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.

scholarly journals Nonexistence of Global Solutions to the Initial Boundary Value Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Changming Song ◽  
Jina Li ◽  
Ran Gao
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Type Equation ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Bond Number ◽  
Singularly Perturbed ◽  
Sixth Order ◽  
Initial Boundary Value

We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bidirectional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The nonexistence of global solution to the initial boundary value problem for the singularly perturbed Boussinesq-type equation is discussed and two examples are given.

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