Chaotic semigroups generated by certain differential operators of order 1

Abstract

We consider the initial value problem of a partial differential equation $\frac{\partial u}{\partial t}=c(x)\frac{\partial u}{\partial x}+g(x,u)$ in some function spaces $X$ on the interval $I$ of the real line. By using the representation formula of the solution to the equation, we define a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ of bounded linear operators on $X$. When $c(x)=\text{γ}x(\text{γ}\in ℝ),g(x,u)=h(x)u(h\in C(I,ℂ))$ and $I$ is $[0,1]$ or $[1,\infty )$, we give sufficient conditions for the semigroup to be chaotic by using the spectral property of its infinitesimal generator. When $c(x)=1$ and $g(x,u)=h(x)u$, we also give sufficient conditions for the semigroup to be chaotic by using the property of an admissible weight function.