Multiplication Semigroups in Variable Exponent Lebesgue Spaces

This paper studies multiplication operators and their associated strongly continuous semigroups acting on variable exponent Lebesgue spaces. We study the abstract Cauchy problem $\dot{u}\left(t\right)=Au\left(t\right)$, $u\left(0\right)=u_0$, in the space $L^{p\left(x\right)}(0,\ell )$ with $\ell >0$, where the generator A is given by the multiplication operator $A=M_q$. Using the modular ${\rho }_{p(·)}\left(u\right)={\int }_0^{\ell }{\left|u\left(x\right)\right|}^{p\left(x\right)}dx$, we establish the fundamental properties of $M_q$, including ${\rho }_{p(·)}$-closedness, density of its domain, and boundedness criteria in terms of the essential range of q.We show that $M_q$ generates a strongly continuous semigroup ${\left(S\left(t\right)\right)}_{t\ge 0}$ given explicitly by $S\left(t\right)=e^{tA}=M_{e^{tq}}$, and we derive modular growth estimates for the semigroup. We also obtain a complete characterization of the spectrum and resolvent of A, showing that $\sigma \left(A\right)=q_{\mathrm{ess}}(0,\ell )$ and $R(\lambda ,A)={(\lambda I-A)}^{-1}=M_{1/(\lambda -q)}$ for $\lambda \notin \sigma \left(A\right)$. The spectral mapping behavior of the associated semigroup is also analyzed, highlighting the validity of the weak spectral mapping theorem and the possible failure of the full spectral identity. As an application, we present a concrete example on $(0,4)$ involving a singular initial datum that does not belong to $L^2(0,4)$ but lies in $L^{p\left(x\right)}(0,4)$ due to a suitable spatial variation of the exponent. The corresponding evolution is explicitly given by $u(t,x)=e^{tq\left(x\right)}f\left(x\right)$ and remains well posed in $L^{p\left(x\right)}(0,4)$ for all $t\ge 0$. This shows that the variable exponent framework can accommodate singular behavior while preserving semigroup dynamics. These results show that multiplication operators provide an explicit model for semigroup theory in variable exponent spaces, connecting modular analysis with pointwise evolution equations.