Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

Abstract

<p style='text-indent:20px;'>Traveling wave solutions for the one-dimensional degenerate parabolic equations are considered. The purpose of this paper is to classify the nonnegative traveling wave solutions including sense of weak solutions of these equations and to present their existence, information about their shape and asymptotic behavior. These are studied by applying the framework that combines Poincaré compactification and classical dynamical systems theory. We also aim to use these results to generalize the results of our previous studies. The key to this is the introduction of a transformation, which overcomes the generalization difficulties faced by these studies.</p>

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

“…Furthermore, it should be noted that the weak traveling wave solution obtained for δ = 1 in Theorem 2 of [14] is partially utilized in the discussion of [3]. As mentioned in [11], Ichida-Matsue-Sakamoto [12] gave a refined asymptotic behavior, which was not obtained in the preceding work [14], by an appropriate asymptotic study and properties of the Lambert W function. Since the discussion process uses the method of blow-up technique, it is necessary to assume 1 < p ∈ N in [14,12] from previous studies [1,7,10].…”

“…Since the discussion process uses the method of blow-up technique, it is necessary to assume 1 < p ∈ N in [14,12] from previous studies [1,7,10]. Therefore, there is no discussion for the case that 1 < p ∈ R. The author [11] uses the transformation…”

“…The key to this argument is that in the classification of traveling wave solutions of (7) obtained by (6), it is not necessary to use the blow-up technique since p appears in the coefficient part instead of the exponent in the calculation process. See [11] for more details. This result generalizes the result in [14,12] to the case of 1 < p ∈ R.…”

“…The methods (in particular, the Poincaré compactification) employed in the above previous studies [14,11] are briefly described. The Poincaré compactification is the key method in this paper.…”

“…The Poincaré compactification is the key method in this paper. See Appendix A and [10,13,14,11,17,18] for details and geometric images. This is one of the compactifications of the original phase space, the embedding of R n into R n+1 in the unit's upper hemisphere.…”