Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions

2010 ◽  
Vol 11 (5) ◽  
pp. 3815-3823 ◽  
Author(s):  
Yuanfei Li ◽  
Yan Liu ◽  
Changhao Lin
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Parabolic Problems ◽  
Nonlinear Parabolic Problems ◽  
Nonlinear Parabolic

scholarly journals Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions

2019 ◽  
Vol 150 (1) ◽  
pp. 475-495 ◽  
Author(s):  
Begoña Barrios ◽  
Maria Medina
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Classical Case ◽  
Parabolic Problems ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Local Version ◽  
Comparison Results ◽  
Non Local

AbstractWe present some comparison results for solutions to certain non-local elliptic and parabolic problems that involve the fractional Laplacian operator and mixed boundary conditions, given by a zero Dirichlet datum on part of the complementary of the domain and zero Neumann data on the rest. These results represent a non-local generalization of a Hopf's lemma for elliptic and parabolic problems with mixed conditions. In particular we prove the non-local version of the results obtained by Dávila and Dávila and Dupaigne for the classical cases= 1 in [23, 24] respectively.

Heterogeneous Elliptic BVPs with a Bifurcation-Continuation Parameter in the Nonlinear Mixed Boundary Conditions

2020 ◽  
Vol 20 (1) ◽  
pp. 31-51
Author(s):  
Santiago Cano-Casanova
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Blow Up ◽  
Mixed Boundary ◽  
Elliptic Boundary ◽  
Global Bifurcation ◽  
Mixed Boundary Conditions ◽  
Elliptic Boundary Value ◽  
New Findings ◽  
Continuation Parameter

AbstractThis article ascertains the global structure of the diagram of positive solutions of a very general class of elliptic boundary value problems with spatial heterogeneities and nonlinear mixed boundary conditions, considering as bifurcation-continuation parameter a certain parameter γ that appears in the boundary conditions. In particular, in this work are obtained, in terms of such a parameter γ, the exact decay rate to zero and blow-up rate to infinity of the continuum of positive solutions of the problem, at the bifurcations from the trivial branch and from infinity. The new findings of this work complement, in some sense, those previously obtained for Robin linear boundary conditions by J. García-Melián, J. D. Rossi and J. C. Sabina de Lis in 2007. The main technical tools used to develop the mathematical analysis carried out in this paper are local and global bifurcation, continuation, comparison and monotonicity techniques and blow-up arguments.

scholarly journals Global and Blow-Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lingling Zhang ◽  
Hui Wang
Keyword(s):  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Gradient Term ◽  
Nonlinear Parabolic Problems ◽  
Nonlinear Parabolic ◽  
Robin Boundary ◽  
Upper Estimate

We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions:(b(u))t=∇·(h(t)k(x)a(u)∇u)+f(x,u,|∇u|2,t), inD×(0,T),(∂u/∂n)+γu=0, on∂D×(0,T),u(x,0)=u0(x)>0, inD¯, whereD⊂RN  (N≥2)is a bounded domain with smooth boundary∂D. Under some appropriate assumption on the functionsf,h,k,b, andaand initial valueu0, we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

scholarly journals Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mauricio Bogoya ◽  
Julio D. Rossi
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Asymptotic Bounds ◽  
Blowing Up ◽  
Blow Up Rate ◽  
Blowing Up Solutions

We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

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