Input-to-State Stability for a Class of One-Dimensional Nonlinear Parabolic PDEs with Nonlinear Boundary Conditions

2020 ◽  
Vol 58 (4) ◽  
pp. 2567-2587 ◽  
Author(s):  
Jun Zheng ◽  
Guchuan Zhu
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Parabolic Pdes ◽  
One Dimensional ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

scholarly journals Continuum of one-sign solutions of one-dimensional Minkowski-curvature problem with nonlinear boundary conditions

2021 ◽  
Author(s):  
Yanqiong Lu ◽  
Zhengqi Jing
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Bifurcation ◽  
Nonlinear Boundary Conditions ◽  
Continuous Functions ◽  
One Dimensional ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
Curvature Problem ◽  
The Continuum

In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.

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