Global and blow-up solutions for a heat equation with variable reaction

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This article discusses the existence of global and blow-up solutions for the semilinear heat equation with a variable exponent. The equation is given by u t − Δ u = h ( t ) f ( u ) p ( x ) in Ω × ( 0 , T ) with zero Dirichlet boundary condition and initial data in C 0 ( Ω ) . Our analysis covers both bounded and unbounded domains, p ( ⋅ ) is a continuous function in Ω with 0 < p − ≤ p ( x ) ≤ p + , h ∈ C ( 0 , ∞ ) and f ∈ C [ 0 , ∞ ) . Our findings have significant implications as they improve upon the blow-up result discovered by Castillo and Loayza in Comput. Math. App. 2017;74(3):351–359 when f ( u ) = u .