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In this paper, we give a formal solution of some nonlinear evolution equations. By the formal solution, we can obtain the blow-up solution of the heat equations, even in the supercritical case. [ABSTRACT FROM AUTHOR]

The phylum Acanthocephala is an important group of parasites with more than 1,300 species parasitizing intestine of all major vertebrate groups. However, our present knowledge of the mitochondrial genomes of Acanthocephala remains very limited. In the present study, we sequenced and annotated the complete mitochondrial genome of Acanthogyrus (Acanthosentis) bilaspurensis (Gyracanthocephala: Quadrigyridae) for the first time based on the specimens recovered from the intestine of common carp Cyprinus carpio Linnaeus (Cyprinidae) in Pakistan. The mitochondrial genome of A. bilaspurensis is 13,360 bp in size and contains 36 genes, representing the smallest mitogenome of acanthocephalans reported so far. The mitogenome of A. bilaspurensis also has the lowest level of overall A+T contents (59.3%) in the mitogenomes of Eoacanthocephala, and the non-coding region 3 (NCR3) lies between trn S2 and trn I, which is different from all of the other acanthocephalan species. Phylogenetic analyses based on concatenating the amino acid sequences of 12 protein-coding genes using maximum likelihood (ML) and Bayesian inference (BI) methods revealed that the family Pseudoacanthocephalidae is a sister to the Arhythmacanthidae rather than the Cavisomatidae, and the families Rhadinorhynchidae and Cavisomatidae showed sister relationships. [ABSTRACT FROM AUTHOR]