Fujita exponents for an inhomogeneous parabolic equation with variable coefficients. (10th March 2022)

• Record Type: Journal Article
• Authors: Sun, Xizheng; Liu, Bingchen; Li, Fengjie
• Is Part Of: Mathematical methods in the applied sciences. Volume 45:Number 11(2022)
• Page Start: 7058
• Page End: 7071
• Publication Date: 2022-03-10
• Subjects: blow‐up -- Cauchy problem -- Fujita exponent -- inhomogeneous parabolic equation -- variable coefficient; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8224
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This paper deals with a Cauchy problem of the inhomogeneous parabolic equation u t = Δ u + 〈 x 〉 γ u p + t σ w ( x ) in ℝ N × ( 0, T ), where constants γ > 0, p > 1, and σ > − 1 . The Japanese brackets 〈 x 〉 γ : = 1 + | x | 2 γ ; w ( ≥, ≢ 0 ) and the initial data are continuous functions in ℝ N . We determine the Fujita exponent for global and non‐global solutions of the problem, depending strictly on N, γ and σ, which complete the ones for the nonnegative solutions in J. Math. Anal. Appl. 251 (2000) 624–648 for N = 1, 2 . It is so interesting that the inhomogeneous term leads to the discontinuity of this critical exponent with respect to σ at zero.