Supersolutions for a class of semilinear heat equations
By James C. Robinson, M. Sierżęga. 2011
49 citations recorded by the source corpus.
A semilinear heat equation $$u_{t}=\Delta u+f(u)$$ with nonnegative measurable initial data is considered under the assumption that $$f$$ is nonnegative and nondecreasing and $$\Omega \subseteq \mathbb R ^{n}$$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is then applied to the model case $$f(s)=s^{p}$$ with initial data in $$L^{q}(\Omega )$$, for which an extension of the monotonicity-based existence argument is offered for the critical case ($$n(p-1)/2=q>1$$) in all dimensions. New sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical ($$n(p-1)/2 q>1$$) range of parameters. Some possible generalisations of the method to a broader class of equations are discussed.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/75c993c60d1c7248f2bcad9f1f460c12851091e4
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In Applied Mathematics
In Control and Systems Engineering
In Mathematical Physics
Citations in the corpus
- Existence and nonexistence of solutions for the heat equation with a superlinear source term (2016, background): “See [1], [3], [4], [6]–[12], [15], [18], [20]–[26] for existence and nonexistence of solutions for nonlinear parabolic equations and qualitative properties of solutions.”
- Existence of solutions for a fractional semilinear parabolic equation with singular initial data (2016, methodology): “We modify the arguments in [11, 17] and prove the following two theorems.”