Fujita exponents for an inhomogeneous parabolic equation with variable coefficients. (10th March 2022) - Sun, Xizheng; Liu, Bingchen; Li, Fengjie
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.