Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions

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Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems (2004, background): “The proof in [7] uses heat kernel estimates.”
Singularity and decay estimates in superlinear problems via liouville-type theorems. Part II: Parabolic equations (2007, background): “Earlier results on universal bounds for bounded domains were proved in [12] [31].”
A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains (2007, methodology): “As it turns out positive very weak solutions are very useful in the study of parabolic blow-up, see Section 4. The main tool of Quittner, Souplet were sharp estimates of the heat semigroup obtained earlier by Fila, Souplet and Weissler [ 6 ].”
Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level (2005, background): “For positive global solutions, stronger universal bounds are available, see [12,35,37].”