Discretely self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field for large discretely self-similar initial data are constructed in this note, extending the construction of Brandolese and Karch on self-similar solutions. It follows the approach of Bradshaw and Tsai and finds an explicit a priori bound for the deviation from suitably revised profiles in similarity variables.
We define a hypergeometric series in m variables with p+(p-1)m parameters, which reduces to the generalized hypergeometric series _pF_p-1 when m=1, and to Lauricella's hypergeometric series F_C in m variables when p=2. We give a system of hypergeometric differential equations annihilating the series. Under some non-integral conditions on parameters, we give an Euler type integral representation of the series, and linearly independent p^m solutions to this system around a point near to the origin. We show that this system is of rank p^m, and determine its singular locus.
This study addresses the asymptotic behavior of solutions of the dynamic Dixit-Stiglitz-Krugman model. A sufficient condition is obtained for solutions of this model to converge to one region as time increases. Examples are provided to illustrate the main results for the three-region model.
In this paper, we find new regularity criteria on the magnetohydrodynamic (MHD) system in terms of partial components of the velocity field and the magnetic field in Lorentz space. Our aim is to refine and extend the criteria of the Lebesgue space in previous results to those of a larger Lorentz space.
In this study, we consider a fractional-order predator-prey model with a Holling-type functional response and stage structure for the prey. This study involved analyzing the existence and uniqueness of nonnegative solutions of the system. Conditions for the local asymptotic stability of the equilibria are established, and controllers are designed for the global asymptotic stability of the interior equilibrium. Furthermore, numerical simulations are performed to illustrate the results.
In 1993, Gerard-Tahara introduced the Fuchs (Gerard-Tahara) type partial differential equation, and they determined the structure of holomorphic and singular solutions provided that the characteristic exponents satisfy some conditions. In this paper the author shows existences of holomorphic and singular solutions of difference-differential equations where the time variable derivative is replaced by q-difference operator.
We consider the asymptotic behavior of solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity. It is known that the leading term of the asymptotic profile for the solution to this problem is the Gaussian. Moreover, by analyzing the corresponding integral equation, the higher-order asymptotic expansion for the solution to the linear part and the first asymptotic profile for the Duhamel term have already been obtained. In this paper, we construct the second asymptotic profile for the Duhamel term and give the more detailed higher-order asymptotic expansion of the solutions, which generalizes the previous works. Furthermore, we emphasize that the newly obtained higher-order asymptotic profiles have a good structure in the sense of satisfying the parabolic self-similarity.
In this paper, we analyze metrical approximations of functions F : Lambda x X -> Y by trigonometric polynomials and p-periodic type functions, where Phi not equal Lambda subset of R-n, X and Y are complex Banach spaces, and p is a general binary relation on Y. Besides the classical concept, we analyze Stepanov, Weyl, Besicovitch and Doss generalized approaches to metrical approximations. We clarify many structural properties of introduced spaces of functions and provide some illustrative applications to the abstract PDEs.
We prove that Sobolev norms of solutions to time-dependent Schrodinger equations for multi-particle systems interacting via time-dependent two body potentials are bounded in time if certain Sobolev norms of the potentials are small uniformly in time. The proof uses the scattering theory in the extended phase space which proves that all particles scatter freely in the remote past and far future.
We consider the Cauchy problem for a quadratic derivative nonlinear Schrodinger equation whose nonlinearity is a linear combination of partial derivative(x)(u(2)) and partial derivative(x)(|u|(2)). We prove the local well-posedness in the L-2-based Sobolev space H-s (R) for s >= 0 with bounded primitives. Moreover, we prove the global well-posedness in H-s (R) for s >= 1 and a special case of the coefficients of nonlinearities.
For a general solution of the third Painlev\'e equation of complete type we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Jacobi sn-function in cheese-like strips along generic directions. The expression is derived by using isomonodromy deformation of a linear system governed by the third Painlev\'e equation of this type. In our calculation of the WKB analysis, the treated Stokes curve ranges on both upper and lower sheets of the two sheeted Riemann surface.
A new approach is used to obtain a global solvability criterion for matrix Riccati equations. It is shown that the obtained result is an extension of a result derived from a comparison theorem for matrix Riccati equations. Two corollaries were drawn from the obtained result as well.
We study the existence of a periodic solution for a differential equation with distributed delay. It is shown that, for a class of distributed delay diferential quations, a symmetric period 2 solution, where the period is twice the maximum delay, is given as a periodic solution of a Hamiltonian system of ordinary differential equations. Proof of the idea is based on (Kaplan Yorke, 1974, J. Math. Anal. Appl.) for a discrete delay differential equation with an odd nonlinear function. To illustrate the results, we present distributed delay differential equations that have periodic solutions expressed in terms of the Jacobi elliptic functions.
We consider the ill-posedness issue for the cubic nonlinear heat equation and prove norm inflation with infinite loss of regularity in the Ho & uml;lder-Besov space cPs= Bsy, yfor s <= -2/3. In particular, our result includes the subcritical range -1 < s <= -2/3, which is above the scaling critical regularity s = -1 with respect to the Ho & uml;lder-Besov scale. In view of the well-posedness result in b(s), s > -2/3, our illposedness result is sharp.
In this paper, we consider the higher order parabolic Schrodinger type operator with nonnegative potentials. We assume that the nonnegative potential belongs to both the reverse Holder class and the Gaussian class. Using the pointwise estimates of Hardy-Littlewood maximal operator, we establish L-p boundedness of Riesz transforms associated with the higher order parabolic Schrodinger type operator.
We establish necessary and sufficient conditions for the existence of asymptotic periodic solutions of delay evolution equations on half line. These conditions are determined by the spectrum of forcing term and spectrum of evolution semigroups associated with the equations. Our goal is to extend on Massera's theorem by recalling a new concept of solutions, namely, asymptotic solutions for evolution equations in Banach spaces, which have potential applications to partial differential equations as well as abstract functional differential equations.
It is well known that a time delay can cause the instability of the equilibrium. In the case of pathogen dynamics models, the absorption effect (when pathogens invade target cells, they are absorbed into cells and disappear from blood) also can cause the instability. Besides the two effects above, introducing an immune response variable into a pathogen dynamics model can cause the instability. This paper shows, for the models lacking one or more of the three effects, the positive equilibrium is always stable. Moreover, it shows that, for the model incorporating all the three effects, the positive equilibrium can be unstable.
In this paper, we consider the Cauchy problem for the linear dispersive equations. We establish the probabilistic radial Strichartz estimates for a randomization preserving the radial symmetry. As an application, we prove almost sure local well-posedness along with small data global existence and scattering for the radial mass-critical fractional Schrodinger equation.
