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This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the uniform attractors for 3D MHD equations with nonlinear damping. Some uniform estimates for strong solutions are established. Furthermore, we prove that 3D damped equations have an $ (\mathbb {V},\mathbb {V}) $ (V , V) -uniform attractor and an $ (\mathbb {V},\mathbf{H}^{2}) $ (V , H 2) -uniform attractor, and the $ (\mathbb {V},\mathbb {V}) $ (V , V) -uniform attractor is actually the $ (\mathbb {V},\mathbf{H}^{2}) $ (V , H 2) -uniform attractor. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiple solutions of parametric quasi-linear systems of the gradient-type on the Sierpiński gasket arising in physical problems leading to nonlinear models involving reaction-diffusion equations, in problems on elastic fractal media or fluid flow through fractal regions. By using some critical point theorems, we give some new results to establish one, two and three weak solutions for our system. Finally, we give four examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, the existence and the uniqueness of an evolutionary anisotropic $ p_i(x) $ p i (x) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $ L^{\infty } $ L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $ L^{\infty } $ L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $ p(x)- $ p (x) − Laplace equation with convection term via $ L^{\infty }- $ L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) $$\begin{align*} \left \{ \begin{array}{@{}l@{}} \displaystyle -\left(a+b\int_{R^{N}}|\nabla u|^{2}\,{\rm d}x\right)\Delta u + V(x)u =P(x)|u|^{2^*-2}u+\mu|u|^{q-2}u, \ {\rm in}\ \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N) \end{array} \right. \end{align*}$$ { − (a + b ∫ R N | ∇u | 2 d x) Δu + V (x) u = P (x) | u | 2 ∗ − 2 u + μ | u | q − 2 u , in R N , u ∈ H 1 (R N) where $ a, b, \mu \gt 0 $ a , b , μ > 0 , $ N\geq 3 $ N ≥ 3 , $ \max \{2^*-1, 2\} \lt q \lt 2^*=\frac {2N}{N-2} $ max { 2 ∗ − 1 , 2 } < q < 2 ∗ = 2 N N − 2 , $ V(x) \gt 0 $ V (x) > 0 and $ P(x)\geq 0 $ P (x) ≥ 0 are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P). [ABSTRACT FROM AUTHOR]

In the paper we provide systematic approaches to study existence and feedback control for a new evolutionary equation involving pseudomonotone operators. We first establish several existence results for the evolutionary equation by exploiting the Rothe method and using a surjectivity result for multivalued pseudomonotone operators. Then we show the existence of feasible pairs for the feedback control problem by assuming some sufficient conditions. Moreover, the existence of solutions to an evolutionary hemivariational inequality is presented. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

Parameter identification problems in PDEs are special class of nonlinear inverse problems which has many applications in science and technology. One such application is the Electrical Impedance Tomography (EIT) problem. Although many methods are available in literature to tackle nonlinear problems, the computation of Fréchet derivative is often a bottle neck for deriving the solution. Moreover, many assumptions are required to establish the convergence of such methods. In this paper, we propose a modified form of Levenberg–Marquardt scheme which does not require the knowledge of exact Fréchet derivative, instead, uses an approximate form of it and at the same time, no additional assumptions are required to establish the convergence of the scheme. We illustrate the theoretical result through numerical examples. In order to ensure that the proposed scheme can be applied to practical problems, we have applied the scheme to EIT problem and the reconstruction process clearly demonstrates that the method can be successfully applied to practical problems. [ABSTRACT FROM AUTHOR]

In this paper, an initial value problem for a nonlinear time-fractional Schrödinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions. [ABSTRACT FROM AUTHOR]

We consider periodic solutions of the following nonlinear equation associated with the fractional Laplacian \[ (-\Delta)^s {u}(x)+V(x){u}(x)= {f(x,u)}, \quad x\in \mathbb{R}, \] (− Δ) s u (x) + V (x) u (x) = f (x , u) , x ∈ R , where $ V(x)\in C(\mathbb R) $ V (x) ∈ C (R) is T-periodic and 0 belongs to a spectral gap of the operator $ (-\Delta)^s+V $ (− Δ) s + V. We obtain the existence of a periodic solution by using a generalized linking theorem. [ABSTRACT FROM AUTHOR]

We study the asymptotic expansion of transmission eigenvalues for anisotropic thin layers. We establish a rigorous second-order expansion for simple transmission eigenvalues with respect to the thickness of the layer. The convergence analysis is based on a generalization of Osborn's Theorem to non-linear eigenvalue problems by Moskow [Nonlinear eigenvalue approximation for compact operators. J Math Phys. 2015;56(11):113512]. We also provide formal derivation in more general cases validating the obtained theoretical result. [ABSTRACT FROM AUTHOR]