Heat equations associated to harmonic oscillator with exponential nonlinearity

We investigate the Cauchy problem for a heat equation involving a fractional harmonic oscillator and an exponential nonlinearity:

$$\begin{aligned} \partial _tu + (-\Delta +\varrho |x|^2)^{\beta }u=f(u), \quad (x,t)\in {\mathbb {R}}^d\times (0,\infty ), \end{aligned}$$

where \(\varrho \ge 0,~\beta >0\) and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) exhibits exponential growth at infinity, with \(f(0)=0.\) We establish local well-posedness within the appropriate Orlicz spaces. Through the examination of small initial data in suitable Orlicz spaces, we obtain the existence of global weak-mild solutions. Additionally, precise decay estimates are presented for large time, indicating that the decay rate is influenced by the nonlinearity’s behavior near the origin. Moreover, we highlight that the existence of local nonnegative classical solutions is no longer guaranteed when certain nonnegative initial data are considered within the appropriate Orlicz space.