Some qualitative analysis for a parabolic equation with critical exponential nonlinearity (Q6624898)

This paper is concerned the blow-up phenomena for solutions of a parabolic equation with a critical exponential source and arbitrary positive initial energy of the form\N\[ \Nv_t - \Delta v + v = \lambda f(v), \quad x \in \mathbb{R}^2, \ t > 0, \N\]\Nwith initial condition\N\[\Nv(x, 0) = v_0(x), \quad x \in \mathbb{R}^2,\N\]\Nwhere \( v_0 \in H^1(\mathbb{R}^2)\) and the parameter \(\lambda\) satisfies\N\[ \N0 < \lambda < \frac{1}{2\alpha_0}. \N\]\N\NThe study offers a valuable contribution to the analysis of parabolic equations with critical exponential nonlinearity and arbitrary positive initial energy.\N\NThe examination of the effects of the parameter \(\lambda\) on the behavior of solutions is a notable aspect of this work where they place their results in the context of various applications in plasma physics, nonlinear optics, and astrophysics, as well as their relevance to combustion theory and the study of Riemann metrics. This extension is noteworthy and enhances the understanding of the dynamics of solutions to parabolic equations with critical exponential sources.