The Cauchy problem for an equation of Sobolev type with power non-linearity

Cauchy problems of pseudo-parabolic equations with inhomogeneous terms

“…here F −1 ξ→x denotes the inverse Fourier transform, and the operator B k = (I − kΔ) −1 is given by the Bessel-Macdonald kernel [17] …”

“…Here we use the integral representation and contraction-mapping principle inspired by [17] to prove the global existence results for solutions of the Cauchy problem (1.3). Define the function space…”

“…Thus, the equation of the form (1.1) is also called a Sobolev type equation. Since the last century, pseudo-parabolic equations have been studied in different aspects, such as the integral representations of solutions [12], long-time behavior of solutions [17], Riemann problem and Riemann-Hilbert problem [8], nonlocal boundary value problems [4], and periodic problems [6,18,24,25]. Cao et al [5] investigated the blow-up theorems of the Cauchy problem…”

“…Furthermore, in the proof of Theorem 1(i) in [39], we remove the restrict ϕ > 0 in one dimension and the assumption ϕ ∈ I α for α > α * , and so we improve part of results in paper [39]. As to the global existence of solutions, due to simultaneous appearance of kΔu t and inhomogeneous term |x| σ , we have to develop some techniques to overcome technical difficulties, and so our proofs are more complicated than the proofs for the case σ = 0 [5,17,39]. From our results, it is worth to point out that, like parabolic equations, the asymptotic behavior of solutions to (1.3) is seriously affected by the inhomogeneous term |x| σ .…”

Second critical exponent and life span for pseudo-parabolic equation

“…In resent years considerable attentions have been paid to the study of pseudo-parabolic equations. Refer to [10,15,16,19,22,23] and references therein, where existence and uniqueness of solutions, regularity of solutions, optimal control problems, periodic problems, etc. were considered for various pseudo-parabolic equations.…”