Non-simultaneous Singularity For Two Classes Of Nonlinear Parabolic Equations With Variable Exponents

This paper deals with asymptotic properties for two classes of nonlinear parabolic equations with nonstandard growth conditions. There are two subjects included: the quenching phenomena and the blowup phenomena of singular solutions for two classes of parabolic equations with variable exponents respectively. The singular classification and asymptotic properties are obtained, represented by using multiple variable exponents.We use the traditional definitions of quenching and blow-up. After obtaining the conditions of the occurring of the singularities, we show the complete and optimal classifications of simultaneous and non-simultaneous singularities:(i)there exist initial data such that non-simultaneous singularities occur;(ii)simultaneous singularities occur for any initial data;(iii)the coexistence of simultaneous and non-simultaneous singularities;(iv)non-simultaneous singularities occur for any initial data. All of the classifications and asymptotic properties are described by the maxima of the variable exponents.