On the Existence of Weak-Solutions to an n-Dimensional Stefan Problem with Nonlinear Boundary Conditions

1980 ◽  
Vol 11 (4) ◽  
pp. 632-645 ◽  
Author(s):  
J. R. Cannon ◽  
Emmanuele DiBenedetto
Keyword(s):  
Boundary Conditions ◽  
Stefan Problem ◽  
Weak Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Of Weak Solutions

scholarly journals GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR STRONGLY DAMPED WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS AND BALANCED POTENTIALS

2019 ◽  
Vol 99 (03) ◽  
pp. 432-444
Author(s):  
JOSEPH L. SHOMBERG
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Weak Solutions ◽  
Wave Equations ◽  
Polynomial Growth ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Dynamic Boundary Conditions ◽  
Existence Of Weak Solutions ◽  
Strongly Damped

We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. The associated linear operator is $(-\unicode[STIX]{x1D6E5}_{W})^{\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{t}u$ , where $\unicode[STIX]{x1D703}\in [\frac{1}{2},1)$ and $\unicode[STIX]{x1D6E5}_{W}$ is the Wentzell–Laplacian. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth of each potential, as well as mixed dissipative/antidissipative behaviour.

scholarly journals Infinitely Many Weak Solutions of thep-Laplacian Equation with Nonlinear Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Feng-Yun Lu ◽  
Gui-Qian Deng
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Nonlinear Boundary ◽  
Smooth Boundary ◽  
Nonlinear Boundary Conditions ◽  
Fountain Theorem ◽  
Laplacian Equation ◽  
Variant Fountain Theorem

We study the followingp-Laplacian equation with nonlinear boundary conditions:-Δpu+μ(x)|u|p-2u=f(x,u)+g(x,u),  x∈Ω,|∇u|p-2∂u/∂n=η|u|p-2uandx∈∂Ω,  whereΩis a bounded domain inℝNwith smooth boundary∂Ω. We prove that the equation has infinitely many weak solutions by using the variant fountain theorem due to Zou (2001) andf,gdo not need to satisfy the(P.S)or(P.S*)condition.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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