scholarly journals Positive Solutions for a Singular Elliptic Equation Arising in a Theory of Thermal Explosion

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2173
Author(s):  
Song-Yue Yu ◽  
Baoqiang Yan
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Thermal Explosion ◽  
Comparison Principle ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Nonlinear Boundary Conditions ◽  
Uniqueness Of The Solution ◽  
Thermal Explosion Model

In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case K(x)>0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)<0 is proved, as well as the monotonicity of the solutions on parameter λ. Finally, we obtain some new results for the existence of solutions, and the dependence on the λ for the case K(x) is sign-changing.

scholarly journals The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

2020 ◽  
Vol 12 (1) ◽  
pp. 173-188
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M.I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
Mixed Boundary Conditions ◽  
Multipoint Problem ◽  
Uniqueness Of The Solution

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

scholarly journals Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

2017 ◽  
Vol 147 (3) ◽  
pp. 649-671
Author(s):  
Nsoki Mavinga ◽  
Rosa Pardo
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
Nonlinear Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotically Linear ◽  
Bifurcation From Infinity

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

scholarly journals INFINITE RESONANT SOLUTIONS AND TURNING POINTS IN A PROBLEM WITH UNBOUNDED BIFURCATION

2010 ◽  
Vol 20 (09) ◽  
pp. 2885-2896 ◽  
Author(s):  
J. M. ARRIETA ◽  
R. PARDO ◽  
A. RODRÍGUEZ-BERNAL
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Infinite Number ◽  
Nonlinear Boundary ◽  
Turning Points ◽  
Nonlinear Boundary Conditions ◽  
Steklov Eigenvalue

We consider an elliptic equation -Δu + u = 0 with nonlinear boundary conditions ∂u/∂n = λu + g(λ, x, u), where (g(λ, x, s))/s → 0, as |s| → ∞. In [Arrieta et al., 2007, 2009] the authors proved the existence of unbounded branches of solutions near a Steklov eigenvalue of odd multiplicity and, among other things, provided tools to decide whether the branch is subcritical or supercritical. In this work, we give conditions on the nonlinearity, guaranteeing the existence of a bifurcating branch which is neither subcritical nor supercritical, having an infinite number of turning points and an infinite number of resonant solutions.

A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Oleg A. Repin ◽  
Svetlana K. Kumykova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Type ◽  
Original Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Fractional Differentiation ◽  
Uniqueness Of Solutions ◽  
Uniqueness Of The Solution ◽  
Equation Of Mixed Type

AbstractThe paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.

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