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This paper deals with a Cauchy problem of the inhomogeneous parabolic equation ut=Δu+〈x〉γup+tσw(x) in ℝN×(0,T), where constants γ>0,p>1, and σ>−1. The Japanese brackets 〈x〉γ:=1+|x|2γ; w(≥,≢0) and the initial data are continuous functions in ℝN. 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. [ABSTRACT FROM AUTHOR]

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation u t + | D x | α ∂ x u + ∂ x − 1 ∂ y 2 u + 1 2 ∂ x (u 2) = 0 , α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces H s 1 , s 2 ( R 2) with s 1 > − α − 1 4 and s 2 ≥ 0. Second, we prove that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − (α − 1) (3 α − 4) 4 (5 α + 3) if 4 ≤ α ≤ 5. Finally, we show that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − α (3 α − 4) 4 (5 α + 4) if α > 5. Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl.79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl.90 (2008) 338–352]. [ABSTRACT FROM AUTHOR]

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci.15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to η , in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing L t 2 L x 2 -norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition. [ABSTRACT FROM AUTHOR]

This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity ‖ ρ 0 ‖ L ∞ + ‖ b 0 ‖ L 3 is suitably small and the viscosity coefficients satisfy 3 μ > λ . Here, the initial velocity and initial temperature could be large. The assumption on the initial density does not exclude that the initial density may vanish in a subset of R 3 and that it can be of a nontrivially compact support. Our result is an extension of the works of Fan and Yu (Nonlinear Anal Real World Appl 10:392–409, 2009) and Li et al. (SIAM J Math Anal 45:1356–1387, 2013), where the local strong solutions in three dimensions and the global strong solutions for isentropic case were obtained, respectively. The analysis is based on some new mathematical techniques and some new useful energy estimates. This paper can be viewed as the first result concerning the global existence of strong solutions with vacuum at infinity in some classes of large data in higher dimension. [ABSTRACT FROM AUTHOR]