Recent results and open problems on parabolic equations with gradient nonlinearities.

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Lower bounds for blow-up time in a nonlinear parabolic problem (2009, background): “5) As indicated in [5] it is well known that if p q the solution will not blow up in finite time.”
Dissipative Gradient Nonlinearities Prevent δ$\delta$ ‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models (2024, unknown): “(We refer also to [44, 45] for surveys and results on the topic.)”
The role of self-similarity in singularities of partial differential equations (2008, background): “where is any bounded, open set in dimension d. Depending on the values of p > 1 and q > 1 singularities of (216) may be regional (u blows up in subsets of of finite measure), or even global (the solution blows-up in the whole domain); see for instance [ 140 ] and references therein.”
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena (2004, background): “The semilinear equation ut − u = u p − λ|∇u|q (p, q > 1, λ > 0) has been studied by several authors (see [27] for a recent survey).”