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In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the double nonlinear porous medium equation. For subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time for the positive solution to the double nonlinear porous medium equation. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of subdiffusion equations with Mittag-Leffler nonlinearity. The comparison principle is proved in a bounded domain. The results on the local existence, global existence, and blow-up of solutions to the initial-boundary value problem are obtained. In addition, the results of local existence and blow-up of solutions to the initial problem on the whole Euclidean space are proven. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow‐up solutions. Moreover, we extend the obtained results to the time‐fractional Fisher–KPP equation on the Heisenberg group. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces. [ABSTRACT FROM AUTHOR]

The Duhamel product for two suitable functions f and g is defined as follows: (f ⊛ g) (x) = d d x ∫ 0 x f (x - t) g (t) d t. We consider the integration operator J, J f (x) = ∫ 0 x f (t) d t , on the Frechet space C ∞ of all infinitely differentiable functions in 0 , 1 and describe in terms of Duhamel operators its commutant. Also, we consider the Duhamel equation φ ⊛ f = g and prove that it has a unique solution if and only if φ (0) ≠ 0 . [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples. [ABSTRACT FROM AUTHOR]

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions. [ABSTRACT FROM AUTHOR]

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity u t − k Δ u t − Δ u = I 0 + γ ( | u | p) + ω (x) , (t , x) ∈ (0 , ∞) × R N , where p > 1 , k ≥ 0 , ω (x) ≠ 0 and I 0 + γ is the left Riemann–Liouville fractional integral of order γ ∈ (0 , 1). Based on the test function method, we have proved the blow-up result for the critical case γ = 0 , p = p c for N ≥ 3 , which answers an open question posed by Zhou (2020), and in particular when k = 0 it improves the result obtained by Bandle et al. (2000). An interesting fact is that in the case γ > 0 , the problem does not admit global solutions for any p > 1 and ∫ R N ω (x) d x > 0. [ABSTRACT FROM AUTHOR]