Existence and Nonexistence of the Solutions to the Cauchy Problem of Quasilinear Parabolic Equation with a Gradient Term

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Abstract

This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.

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Correspondence to Mingjun Zhou.

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Supported by the National Natural Science Foundation of China (Nos. 11925105 and 12001227).

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Zhou, M., Leng, Y. Existence and Nonexistence of the Solutions to the Cauchy Problem of Quasilinear Parabolic Equation with a Gradient Term. Lith Math J (2021). https://doi.org/10.1007/s10986-021-09511-2

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MSC

  • 35K59
  • 35B33
  • 35K65

Keywords

  • critical Fujita exponent
  • quasilinear parabolic equation
  • gradient term