Narrow Results

The Klein–Gordon equation with quadratic nonlinearity is frequently used to model various problems of physics. On the other hand, it was shown that in many situations, the use of fractional derivatives instead of ordinary derivatives provides more realistic models. In this work, a one-dimensional sequential time-fractional Klein–Gordon equation posed on a bounded interval, subject to certain initial and boundary conditions, is investigated. The fractional derivatives are considered in the Caputo sense and a weight function $h>0$ is allowed in front of the quadratic nonlinearity. Namely, we first establish general sufficient conditions for the nonexistence of solutions. Next, some particular cases of weight functions $h$ are discussed. In our approach, we make use of the test function method. This method requires an appropriate choice of a family of test functions. In this paper, we make a judicious choice of test functions taking into consideration the nonlocal property of the Caputo fractional derivative, the initial and boundary conditions, and the geometry of the domain.

In this paper, we consider fractional equations involving the logarithmic Laplacian with indefinite nonlinearities:

We investigate a class of nonlinear Schrödinger equations with a generalized Choquard nonlinearity and fractional diffusion. We obtain regularity, existence, nonexistence, symmetry as well as decays properties.

A class of Erdélyi–Kober fractional differential inequalities with a polynomial nonlinearity and a singular potential function is investigated in this paper. By mean of the test function method, we establish sufficient conditions for the nonexistence of global weak solutions. Some examples are provided to illustrate our obtained results. To the best of our knowledge, the issue of nonexistence of global solutions for fractional differential equations or inequalities, involving Erdélyi–Kober fractional derivatives, was never addressed in the literature.

In this paper, we study the question of the existence and nonexistence of solutions for some fractional equations with variable exponents. This paper generalizes some analog results in the classical fractional one. As we know, there are no previous results on the nonexistence of solutions for nonlinear equations with fractional $p(\cdot ,\cdot )$-Laplacian.

As a basic model, we study the 2mth-order quasilinear parabolic equation of diffusion-absorption type

We consider the Cauchy problem in R^N × R₊ with arbitrary singular initial data u₀ ≠ 0 such that u₀(x) = 0 for any x ≠ 0. We prove that, in the most delicate case p = q and , this Cauchy problem admits the unique trivial solution u(·, t) = 0 for t > 0. For λ < λ₀, such nontrivial very singular solutions are known to exist for some semilinear higher-order models. This extends the well-known result by Brezis and Friedman established in 1983 for the semilinear heat equation with p = q = m = 1.

This paper studies the existence of positive solutions for a class of singular boundary value problems of p-Laplacian. By using Global Continuation Theorem and the fixed point index technique, criteria of the existence, multiplicity and nonexistence of positive solutions are established.

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space $R_{+}^n$ by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on $B_1(0)$ through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.

We consider a system of nonlinear biharmonic wave inequalities posed in an exterior domain of ${ℝ}^N$. Three types of boundary conditions are investigated. For each case, the existence and nonexistence of weak solutions are studied. Our study yields naturally existence and nonexistence results for the corresponding stationary system and equation. Many new results are provided and some open questions are proposed.

In this paper, we investigate various properties (e.g. nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear tri-harmonic equations in ${ℝ}^n$ ($n\ge 2$) and bi-harmonic equations in ${ℝ}^2$ with negative exponents. Such kind of equations arise from conformal geometry.