The solvability of the first initial-boundary problem for parabolic and degenerate parabolic equations in domains with a conical point

2010 ◽  
Vol 201 (7) ◽  
pp. 999-1028 ◽  
Author(s):  
Sergey P Degtyarev
Keyword(s):  
Boundary Problem ◽  
Parabolic Equations ◽  
Initial Boundary ◽  
Conical Point ◽  
Degenerate Parabolic Equations ◽  
Initial Boundary Problem ◽  
Degenerate Parabolic

scholarly journals Fujita blow-up phenomena of solutions for a Dirichlet problem of parabolic equations with space-time coefficients

2021 ◽  
Author(s):  
Jiaqi Liu ◽  
Fengjie Li ◽  
Bingchen Liu
Keyword(s):  
Dirichlet Problem ◽  
Initial Data ◽  
Boundary Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Space Time ◽  
Space Domain ◽  
Initial Boundary Problem

This paper deals with a homogeneous Dirichlet initial-boundary problem of parabolic equations with different space-time coefficients, $$u_t =\Delta u + t^{\sigma_1} u^{\alpha} + \langle x\rangle^{n} v^{p},\quad v_t =\Delta v + \langle x\rangle^{m} u^{q} + t^{\sigma_2} v^{\beta},$$ where the eight exponents are nonnegative constants and $\langle x\rangle$ is the Japanese brackets. We obtain the Fujita exponents of solutions, which are determined by the eight exponents and the dimension of the space domain. Moreover, simultaneous or non-simultaneous blow-up of the two components of blow-up solutions is discussed with or without conditions on the initial data.

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 A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Jiebao Sun ◽  
Dazhi Zhang ◽  
Boying Wu
Keyword(s):  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Volterra Model ◽  
Generalized Solutions ◽  
Degenerate Parabolic Equations ◽  
Volterra System ◽  
Degenerate Parabolic ◽  
Initial Boundary Value ◽  
Time Periodic

We consider a cooperating two-species Lotka-Volterra model of degenerate parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.

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