A solution to an open problem for wave equations with generalized Wentzell boundary conditions
The authors consider the initial boundary value problem for a linear wave equation in one-dimensional space with generalized Wentzell boundary conditions of the form: \[ {\partial^2u\over\partial t^2}=c^2{\partial^2u\over\partial x^2},\qquad 0\leq x\leq1\qquad t\in \mathbb{R}, \] \[ \begin{aligned} u(x,0)&=f(x),\quad{\partial u\over\partial t}(x,0)=g(x),\qquad 0\leq x\leq1,\\ \frac{\partial^2u}{\partial x^2}(j,t)&+\beta_j\frac{\partial u}{\partial x} (j,t)+\gamma_ju(j,t)=0,\qquad j=0,1,\quad t\in\mathbb{R}, \end{aligned} \tag \(*\) \] where \(c>0\) and \(\beta_j,\gamma_j\;(j=0,1)\) are scalar coefficients. The main goal of the article is to establish the well posedness of the above initial boundary value problem without any restrictions on the complex numbers \(\beta_0,\beta_1,\gamma_0\) and \(\gamma_1\). To show this, the authors covert the initial boundary value problem to the initial value problem in a Banach space \(X\) \[ u''(t)=Au(t), \quad t\in \mathbb{R}, \qquad u(0)=f, \qquad u(0)=g, \tag \(**\) \] where \(A\) is the linear operator in \(C([0,1])\) defined by \((Af)(x)= c^2f''(x)\), \(0\leq x\leq 1\), with domain \[ D(A)=\{f\in C^2([0,1]: f''(j)+ \beta_jf'(j)+ \gamma_jf(j)= 0\text{ at }j=0,1\}. \] Next, using the method of semigroup theory, the authors prove that the operator \(A\) is the generator of a strongly consine function on \(C[0,1]\). This implies the well posedness of the problem \((**)\) (basing on the well-know theorems from semigroup theory). As a additional consequence, they prove that for the initial boundary value problem with generalized Wentzell boundary conditions on [0,1] for linear heat equations is well posed in \(C[0,1]\).
scientific article; zbMATH DE number 2018275