Open Journal of Mathematical Analysis (OMA)

We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of \(p\)–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent \(p\), which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.

This article introduces what we term Hardy-Hilbert-Mulholland-type integral inequalities, which unify features of Hardy-Hilbert-type and Mulholland-type integral inequalities. These inequalities are parameterized by an adjustable parameter. The obtained constant factors are derived in singular form involving a logarithmic-tangent expression, and their optimality is discussed in detail. Several new secondary inequalities are also established. Complete proofs are provided, including detailed steps and references to intermediate results.

In this paper, we give the definitions of \(s\)-convex set and \(s\)-convex function on Heisenberg group. And some inequalities of Jensen’s type for this class of mappings are pointed out.

This article concerns the problem on the growth and the oscillation of some differential polynomials generated by solutions of the second order non-homogeneous linear differential equation \[\begin{equation*} f^{\prime \prime }+P\left( z\right) e^{a_{n}z^{n}}f^{\prime }+B\left( z\right) e^{b_{n}z^{n}}f=F\left( z\right) e^{a_{n}z^{n}}, \end{equation*}\] where \(a_{n}\), \(b_{n}\) are complex numbers, \(P\left( z\right)\) \(\left( \not\equiv 0\right)\) is a polynomial, \(B\left( z\right)\) \(\left( \not\equiv 0\right)\) and \(F\left( z\right)\) \(\left( \not\equiv 0\right)\) are entire functions with order less than \(n\). Because of the control of differential equation, we can obtain some estimates of their hyper-order and fixed points.

The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^{n}\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.

This paper studies a natural one-parameter extension of the Hardy-Hilbert integral inequality. The proposed generalization introduces a parameter that interpolates between different forms. This allows us to establish a hierarchy among a family of related double integrals. We provide sharp upper bounds expressed in terms of the integral norms of the functions involved. In doing so, we extend a classical result while maintaining the optimality of the constant in the original inequality.

This article concerns the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation with \(p-\)Laplacian \[\Big(\left|u''(t)\right|^{p-2}u''(t)\Big)''-\omega\Big(\left|u'(t)\right|^{p-2}u'(t)\Big)'+V(t)\left|u(t)\right|^{p-2}u(t)=f(t,u(t)),\] where \(p>1\), \(\omega\) is a constant, \(V\in C(\mathbb{R},\mathbb{R})\) is noncoercive and \(f\in C(\mathbb{R}^{2},\mathbb{R})\) is of subquadratic growth at infinity. Some results are proved using variational methods, the minimization theorem and the generalized Clark’s theorem. Recent results in the literature are extended and improved.

In this paper we find viscosity solutions to a system with two parabolic obstacle-type equations that involve two normalized \(p-\)Laplacian operators. We analyze a two-player zero-sum game played on two boards (with different rules in each board), in which at each board one of the two players has the choice of playing in that board or switching to the other board and then play. We prove that the game has a value and show that these value functions converge uniformly (when a parameter that controls the size of the steps made in the game goes to zero) to a viscosity solution of a system in which one component acts as an obstacle for the other component and vice versa. In this way, we find solutions to the parabolic two-membranes problem.

The paper considers real valued stretched graphons defined on the Lebesgue measure space \(([0,\infty),{ m})\). The topological space of these graph functions is equipped with the core Hopf algebra to assign renormalized values to unbounded stretched graphons.

In this paper, the dynamical behaviors of a stochastic competition system with a saturation effect are analyzed. The existence and uniqueness of globally positive solution are proved in detail, and the sufficient conditions for stochastic permanence, strong persistence in the mean, weak persistence, and extinction are obtained respectively. Then the existence and uniqueness of stationary distribution are also obtained under some appropriate assumptions. Finally, several numerical simulations are provided to justify the analytical results.