Global L∞-estimates and dissipative H2-estimates of solutions for retarded reaction–diffusion equations

This paper is concerned with the retarded reaction–diffusion equation ${\partial }_{t}u-\Delta u=f\left(u\right)+G(t,u_t)+h\left(x\right)$ in a bounded domain. We allow both the nonlinear terms $f$ and $G$ to be supercritical, in which case the solutions may blow up in finite time, making it difficult to obtain global estimates. Here we employ some appropriate structure conditions to deal with this problem. In particular, we establish detailed global $L^{\infty }$-estimates and dissipative $H^2$-estimates for the solutions and further enhance the regularity results.