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A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid–fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a 'non-uniqueness counterexample'. These supplement the preceding 'constructive existence' result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier λ -plane, which exhibit exponential decay for large values of λ . This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an elliptic equation defined on R N , N ≥ 1 , and involving the p (u) - Laplacian. When p (u) = p (u (x)) , x ∈ R N , i.e., when p depends on the variable x ∈ R N (through the unknown solution u), we say that we are dealing with the local case of the problem. In this case the p (u) - Laplacian can be considered as a new class of operators with variable exponents. When p (u) = p (α (u)) where α is a scalar function of the unknown solution u, we say that we are dealing with the nonlocal case of the problem. In the present work, the issue of the existence of nontrivial solution in the both cases is addressed. [ABSTRACT FROM AUTHOR]

Resonant phenomena have been observed and investigated in various situations, such as plasma experiments, the maritime security and the microtubule in cell physiology. In this paper, abundant resonant behaviors are studied for the (3+1)-dimensional BKP-Boussinesq equation. We mainly discuss the resonant two- and three-soliton solutions in the (x, y)-plane and (x, z)-plane. The characteristics are given for the kink soliton waves, including expressions, maximums, minimums and velocities. The kink soliton waves in the (x, y)-plane are parallel, and the fusion or fission may occur. The kink soliton waves in the (x, z)-plane are not parallel and the resonant phenomena among them are more complicated. [ABSTRACT FROM AUTHOR]

In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation F (u , ε) = u t ′ - u xx ′ ′ + u u x ′ + ε u 2 = f (x , t) , where F : Ω → C ([ 0 , π ] × [ 0 , ∞)) , Ω = C 2 ([ 0 , π ] × [ 0 , ∞)) × R , u (0 , t) = u (π , t) = 0 , u (x , 0) = g (x) , and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem. [ABSTRACT FROM AUTHOR]

We apply the theory of infinitesimal transformations for the study of a family of 1+1 fifth-order partial differential equations which have been proposed before for the description of multiple kink solutions. In this analysis we perform a complete classification of the Lie symmetries and of the one-dimensional optimal system. The results are applied for the derivation of similarity solutions and in particular we find travel-wave and scaling solutions. We show that the kink-solution of these equations can be recovered by the use of the Lie symmetries, while new solutions are also derived. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global stability of traveling waves for non-monotone infinite-dimensional lattice differential equations with time delay. By establishing the boundedness estimate of the solution of the perturbation equation, we obtain that, for any initial perturbations around the traveling wave, the noncritical traveling waves are globally stable with the exponential convergence rate t − 1 / α e − μ t (μ > 0 a n d 0 < α ≤ 2) , and the critical traveling waves are globally stable with the algebraic convergence rate t − 1 / α in a weighted Sobolev space. [ABSTRACT FROM AUTHOR]

We apply the theory of Lie symmetries in order to study a fourth-order 1+2 evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the specific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined. [ABSTRACT FROM AUTHOR]

This paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc B 1 ⊂ R 2 and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at ∂ B 1 and under appropriate conditions is constructed. It develops logarithmic singularities for r = 0 , r = ∞ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper. [ABSTRACT FROM AUTHOR]

Let (− Δ p ) s , with 0 < s < 1 < p < ∞ , be the fractional p-Laplacian operator. We prove that the span of p-harmonic functions in B 1 is dense in C k ( B ¯ 1). [ABSTRACT FROM AUTHOR]

In this study, a nonlinear fractional logistic equations is proposed in the context of a modified form of the conformable fractional-order derivative. The analytical and approximate-analytical solutions are obtained for the proposed mathematical model via the method of conformable fractional differential transform method (CFDTM) and all obtained results have been analyzed and compared with homotopy perturbation method (HPM). With plotting the figures, we compared the CFDTM, HPM, and exact solution together and conclude that the CFDTM solution is closer than the exact solution in comparison with the HPM. The methods are illustrated by two examples of FOLE and solution is obtained. All obtained solutions have been analyzed and compared graphically to validate the effiency and applicability of all results. [ABSTRACT FROM AUTHOR]

This article describes the emergence of formal methods in theory of partial differential equations (PDE) in the French school of mathematics through Janet's work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem for the 3D magneto-hydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method. [ABSTRACT FROM AUTHOR]

We consider the Navier–Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard L2 a priori estimates and we consider its regular approximations with the fractional power operator (-PΔ)1+α, α>0 small, where P is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard L2 a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator (-PΔ)s with s>54. Using Dan Henry’s semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small f are shown to be global in time and regular. [ABSTRACT FROM AUTHOR]

This paper is concerned with prescribing the fractional Q-curvature on the unit sphere $$\mathbb {S}^{n}$$ endowed with its standard conformal structure $$g_0$$ , $$n\ge 4$$ . Since the associated variational problem is noncompact, we approach this issue with techniques passed by Abbas Bahri, as the well known theory of critical points at infinity, as well as some lesser known topological invariants that appear here as criteria for existence results. [ABSTRACT FROM AUTHOR]

Our task here is to use a version of the ‘vanishing viscosity technique’ to study the critical 2D Quasi-Geostrophic equation . The present paper extends and specializes the results reported in Dlotko et al. (2015). We treat now in more detail the solutions of the critical problem ( α = 1 2 ); in particular their uniqueness, regularity and long time behavior. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of infinitely many sign-changing solutions to some quasilinear and elliptic equation whose divergence part contains exponential term. Our main idea consists of the use of the perturbation method together with invariant sets of descending flow in the framework of classical critical point theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to the optimal distributed control problem governed by a class of nonlinear dispersive equations with cubic nonlinearity, which contains the famous Novikov equation as special case. We first investigate the existence and uniqueness of the weak solution for the controlled system, and then find an optimal solution for the controlled system with the generalized cost functional. Moreover, by means of the method suggested by A.Ya. Dubovitskii and A.A. Milyutin, we establish the first order necessary optimality condition of optimal control for the controlled system in the fixed final horizon case. [ABSTRACT FROM AUTHOR]

This paper deals at first with a fully integrable evolution system of nonlinear partial differential equations (PDEs) which is a generalization of the classical Heisenberg ferromagnet equation. Then the scalar variant of this system is considered. Looking for solutions of special form, the problem of finding explicit solutions of the above-mentioned equations is reduced to the global solvability of overdetermined real-valued systems of nonlinear PDEs. In many cases particular solutions which are not solitons are expressed by classical functions including some special ones as Jacobi elliptic functions, Legendre elliptic functions, and Weierstrass normal elliptic integrals. A geometrical visualization of several solutions is also proposed. [ABSTRACT FROM AUTHOR]