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In this work, we study the Cauchy problem of Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of two partial differential equations (PDEs): One is a quasi-linear wave equation for the director field representing the crystallization of the nematics, and the other is a parabolic PDE for the velocity field characterizing the liquidity of the material. We extend the work in Chen et al. (2020) [11] for a special case to the general physical setup. The Cauchy problem is shown to have global solutions beyond singularity formation. Among a number of progresses made in this paper, a particular contribution is a systematic treatment of a parabolic PDE with only Hölder continuous diffusion coefficient and rough (maybe unbounded) nonhomogeneous terms. [ABSTRACT FROM AUTHOR]

In this paper, we present some unexpected answers to two problems with longstanding interest. More specifically, we show that a bounded primitive function of an asymptotically almost periodic function from R to a Banach space X is remotely almost periodic if and only if c 0 ⊄ X. This gives a natural analogy of the classical Bohl-Bohr-Amerio-Kadets theorem for asymptotically almost periodic functions. Based on this result, we succeed in extending the classical Loomis theorem in [13] to asymptotically almost periodic case without any ergodic conditions, and apply our Loomis type theorems to abstract Cauchy problems. Moreover, we give an example of inhomogeneous Schrödinger equation with asymptotically almost periodic coefficient, whose solution is not asymptotically almost periodic but remotely almost periodic. This demonstrates that remotely almost periodic functions is the natural class for solutions to some partial differential equations, which is similar to a remarkable phenomenon revealed by Shen and Yi (1998) [20] using dynamical system approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a L ∞ functional derivative estimate for the first spatial derivatives of bounded classical solutions u : R N × [ 0 , T ] → R to the Cauchy problem for scalar second order semi-linear parabolic partial differential equations with a continuous nonlinearity f : R → R and initial data u 0 : R N → R , of the form, max i = 1 , … , N ⁡ ( sup x ∈ R N ⁡ | u x i ( x , t ) | ) ≤ F t ( f , u 0 , u ) ∀ t ∈ [ 0 , T ] . Here F t : A t → R is a functional as defined in §1 and x = ( x 1 , x 2 , … , x n ) ∈ R N . We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence ( f n , 0 , u ( n ) ) , where for each n ∈ N , u ( n ) : R N × [ 0 , T ] → R is a solution to the Cauchy problem with zero initial data and nonlinearity f n : R → R , and for which there exists α > 0 such that max i = 1 , … , N ⁡ ( sup x ∈ R N ⁡ | u x i ( n ) ( x , T ) | ) ≥ α , whilst lim n → ∞ ⁡ ( inf t ∈ [ 0 , T ] ⁡ ( max i = 1 , … , N ⁡ ( sup x ∈ R N ⁡ | u x i ( n ) ( x , t ) | ) − F t ( f n , 0 , u ( n ) ) ) ) = 0 . [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a multidimensional scalar conservation law and construct an outer estimate for the domain of dependence of its Kružkov solution. The estimate can be represented as the controllability set of a specific differential inclusion. In addition, reachable sets of this inclusion provide outer estimates for the support of the wave profiles. Both results follow from a modified version of the classical Kružkov uniqueness theorem, which we also present in the paper. Finally, the results are applied to a control problem consisting in steering a distributed quantity to a given set. [ABSTRACT FROM AUTHOR]

In this paper, we find that the unbounded spaces Y σ ( R N ) ( 0 < σ < 2 m − 1 ) can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted L 1 – L ∞ estimates for the solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial differential equations. Our analysis leads to new integral equations of non-convolution type, which can be solved by classical methods. We derive solutions of these integral equations, which in turn lead to solutions of the associated Cauchy problems. [ABSTRACT FROM AUTHOR]

The paper deals with a second order integro-partial differential equation in R n with a nonlocal, degenerate diffusion term. Nonlocal conditions, such as the Cauchy multipoint and the weighted mean value problem, are investigated. The existence of periodic solutions is also studied. The dynamic is transformed into an abstract setting and the results come from an approximation solvability method. It combines a Schauder degree argument with an Hartman-type inequality and it involves a Scorza-Dragoni type result. The compact embedding of a suitable Sobolev space in the corresponding Lebesgue space is the unique amount of compactness which is needed in this discussion. The solutions are located in bounded sets and they are limits of functions with values in finitely dimensional spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the Cauchy problem of the cubic nonlinear Schrödinger equation with derivative in . This equation was known to be the local well-posedness for (Takaoka, 1999 ), ill-posedness for (Biagioni and Linares, 2001 , etc.) and global well-posedness for (I-team, 2002 ). In this paper, we show that it is global well-posedness in the endpoint space , which remained open previously. The main approach is the third generation I-method combined with a new resonant decomposition technique. The resonant decomposition is applied to control the singularity coming from the resonant interaction. [Copyright &y& Elsevier]

Abstract: In this paper, using the vanishing viscosity method, we construct a solution of the Riemann problem for the system of conservation laws with the initial data This problem admits δ-, -shock wave type solutions, and vacuum states. -Shock is a new type of singular solutions to systems of conservation laws first introduced in [E.Yu., Panov, V.M. Shelkovich, -Shock waves as a new type of solutions to systems of conservation laws, J. Differential Equations 228 (2006) 49–86]. It is a distributional solution of the Riemann problem such that for its second component v may contain Dirac measures, the third component w may contain a linear combination of Dirac measures and their derivatives, while the first component u has bounded variation. Using the above mentioned results, we also solve the δ-shock Cauchy problem for the first two equations of the above system. Since -shocks can be constructed by the vanishing viscosity method, they are “natural” solutions to systems of conservation laws. We describe the formation of the -shocks and the vacuum states from smooth solutions of the parabolic problem. The results of this paper as well as those of the above-mentioned paper show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well. [Copyright &y& Elsevier]

We consider the Cauchy problem for the Green–Naghdi equations with viscosity, for small initial data. It is well-known that adding a second order dissipative term to a hyperbolic system leads to the existence of global smooth solutions, once the hyperbolic system is symmetrizable and the so-called Kawashima–Shizuta condition is satisfied. In a previous work, we have proved that the Green–Naghdi equations can be written in a symmetric form, using the associated Hamiltonian. This system being dispersive, in the sense that it involves third order derivatives, the symmetric form is based on symmetric differential operators. In this paper, we use this structure for an appropriate change of variable to prove that adding viscosity effects through a second order term leads to global existence of smooth solutions, for small data. We also deduce that constant solutions are asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of the unique global classical solution with small initial data to the Cauchy problem of a scalar conservation law with degenerate diffusion by establishing the uniform a priori decay estimates of solutions. In order to compensate the degeneracy on the x 1 direction by the diffusion on other directions, we introduce the frequency decomposition method and obtain the low frequency estimate and the high frequency estimate of the solution by the Green's function method and energy method respectively. [ABSTRACT FROM AUTHOR]

The paper studies the lifespan of solutions to Cauchy problems for nonlinear analytic partial differential equations with small analytic data. It is proved that the lifespan of the solution becomes longer as the initial data become smaller. The dependence of the lifespan on the smallness of the data can be sharply described by the property of the equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for a higher order shallow water type equation u t − u t x x + ∂ x 2 j + 1 u − ∂ x 2 j + 3 u + 3 u u x − 2 u x u x x − u u x x x = 0 , where x ∈ T = R / 2 π and j ∈ N + . Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in H s ( T ) with s ≥ − j − 2 2 for arbitrary initial data. By using the I -method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in H s ( T ) with 2 j + 1 − j 2 2 j + 1 < s ≤ 1 . Our results improve the results of A.A. Himonas, G. Misiolek (Himonas and Misiolek, 1998 [11] , 2000 [12] ). [ABSTRACT FROM AUTHOR]

In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions, assuming that the initial data is small and smooth. We establish the same type of lower bound of the lifespan for the problem as that for the Cauchy problem, despite of the weak decay property of the solution in two space dimensions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the Cauchy problem (CGL) in for complex Ginzburg–Landau equations with Laplacian Δ and nonlinear term multiplied by the complex coefficients and , respectively (, , , ). The global existence of strong solutions to (CGL) is established without any upper restriction on but with some restriction on and . The result corresponds to Ginibre and Velo (1996) which is technically proved by combining convolution (regularizing) methods with compactness (localizing) methods, while our proof here is fairly simplified. The key to our proof is the Cauchy problem (CGL) R which is (CGL) with Δ replaced with , where (), (). The solvability of (CGL) R is a direct consequence of Okazawa and Yokota (2002) . Taking the limit of global strong solutions to (CGL) R as yields a global strong solution to (CGL). The result gives also an unbounded version of Okazawa and Yokota (2002) for the initial–boundary value problem on bounded domains. [Copyright &y& Elsevier]

Abstract: The classical Duhamel principle, established nearly 200 years ago by Jean-Marie-Constant Duhamel, reduces the Cauchy problem for an inhomogeneous partial differential equation to the Cauchy problem for the corresponding homogeneous equation. In this paper we generalize this famous principle to a wide class of fractional order differential-operator equations. [Copyright &y& Elsevier]

Abstract: Recently, Novikov found a new integrable equation (we call it the Novikov equation in this paper), which has nonlinear terms that are cubic, rather than quadratic, and admits peaked soliton solutions (peakons). Firstly, we prove that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces (which generalize the Sobolev spaces ) with the critical index . Then, well-posedness in with , is also established by applying Kato''s semigroup theory. Finally, we present two results on the persistence properties of the strong solution for the Novikov equation. [Copyright &y& Elsevier]

Abstract: In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of Coron et al. (2010) , the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in . We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the -sense with , the difference between the actual out-flux and a forecast demand over a fixed time period. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation admits a unique global generalized solution in , a unique global classical solution and asymptotic behavior of the solution. We also prove that the Cauchy problem for the equation has a unique global generalized solution in , a unique global classical solution and asymptotic behavior of the solution. [Copyright &y& Elsevier]

Abstract: In this paper, we consider linear nonconservative Cauchy systems with discontinuous coefficients across the noncharacteristic hypersurface . For the sake of simplicity, we restrict ourselves to piecewise constant hyperbolic operators of the form with , where . Under assumptions, incorporating a sharp spectral stability assumption, we prove that a unique solution is successfully singled out by a vanishing viscosity approach. Due to our framework, which includes systems with expansive discontinuities of the coefficient, the selected small viscosity solution satisfies an unusual hyperbolic problem, which is well-posed even though it does not satisfy, in general, a Uniform Lopatinski Condition. In addition, based on a detailed analysis of our stability assumption, explicit examples of systems checking our assumptions are given. Our result is new and contains both a stability result and a description of the boundary layers forming, at any order. Two kinds of boundary layers form, each polarized on specific linear subspaces in direct sum. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Cauchy problem for the generalized finite-depth-fluid equation , where , k is an integer that is larger than 4. We obtain that it is globally wellposed if and the initial data in are sufficiently small, where and . Moreover, we show that its solution will converge to those of the generalized BO and KdV equations as the depth parameter and , respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we study the asymptotic decay rates to the planar rarefaction waves to the Cauchy problem for a hyperbolic–elliptic coupled system called as a model system of the radiating gas in () if the initial perturbations corresponding to the planar rarefaction waves are sufficiently small in () (). The analysis is based on the -energy method and several special interpolation inequalities. [Copyright &y& Elsevier]

Abstract: The paper studies the longtime behavior of the Kirchhoff type equation with strong damping on . It shows that the related continuous semigroup possesses a global attractor which is connected and has finite fractal and Hausdorff dimension. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem of the Cahn–Hilliard equation First, we construct a local smooth solution to the above Cauchy problem, then by combining some a priori estimates, Sobolev''s embedding theorem and the continuity argument, the local smooth solution is extended step by step to all provided that the smooth nonlinear function satisfies a certain local growth condition at some fixed point and that is suitably small. Secondly, we show that the global smooth solution satisfies the following temporal decay estimates: Here , is a constant depending on τ and is any positive constant which can be chosen sufficiently small. At last, we show that, under a strong assumption on the growth of the nonlinear function at , the asymptotics of solutions of the above Cauchy problem is described by . Here , . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes equations effected by external force of general form in . [Copyright &y& Elsevier]

Abstract: This paper considers the Cauchy problem for a conservation law with a variable unilateral constraint, its motivation being, for instance, the modeling of a toll gate along a highway. This problem is solved by means of nonclassical shocks and its well posedness is proved. Then, the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with a Cauchy problem for a semilinear heat equation with and . We show that if and , then for each positive integer there exist , , with and a global -solution u of (P) such that u is a regular solution of (P) for and blows up at for . [Copyright &y& Elsevier]

Abstract: In this paper we derive the following two properties: the first one is a precise representation of WKB solution to the Cauchy problem of a linear wave equation with a variable coefficient with respect to time, and the second one is the global solvability for Kirchhoff equation in some special classes of nonreal-analytic functions, which is proved by grace of the first property. [Copyright &y& Elsevier]

Abstract: This paper is contributed to the Cauchy problem with initial function . The monotonicity and stability of the positive radial steady states, which are positive solutions of , are discussed with , and φ in weighted Banach spaces. [Copyright &y& Elsevier]

Abstract: A concept of a new type of singular solutions to systems of conservation laws is introduced. It is so-called -shock wave, where is nth derivative of the Dirac delta function (). In this paper the case is studied in details. We introduce a definition of -shock wave type solution for the system Within the framework of this definition, the Rankine–Hugoniot conditions for -shock are derived and analyzed from geometrical point of view. We prove -shock balance relations connected with area transportation. Finally, a solitary -shock wave type solution to the Cauchy problem of the system of conservation laws , , with piecewise continuous initial data is constructed. These results first show that solutions of systems of conservation laws can develop not only Dirac measures (as in the case of δ-shocks) but their derivatives as well. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Klein–Gordon–Schrödinger system with quadratic (Yakuwa) coupling and cubic autointeractions in , and prove the existence and uniqueness of global solution for rough data. The techniques to be used are adapted from a general scheme originally introduced by J. Bourgain to split the data into the low and high frequency parts. [Copyright &y& Elsevier]

Abstract: In the present paper we explain new Levi conditions of type for second-order hyperbolic Cauchy problems. Our goal is to explain the special influence of oscillations in the coefficients. It turns out that such oscillations have an essential influence coupled with the asymptotic behavior of characteristics around multiple points. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Cauchy problem of the long-wave–short-wave resonance equations. By making use of a Strichartz-type inequality for the solutions, decomposing suitably the solution semigroup into a decay parts and a more regular parts, and ruling out the “vanishing” and “dichotomy” of the solutions, we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions. [Copyright &y& Elsevier]

Abstract: This paper concerns the global existence and the large time behavior of strong and classical solutions to the two-dimensional (2D) Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the 2D Stokes approximation equations for the compressible flows together with the space-periodicity boundary condition or the no-stick boundary condition or Cauchy problem for arbitrarily large initial data. First, we prove that the density is bounded from above independent of time in all these cases. Secondly, we show that for the space-periodicity boundary condition or the no-stick boundary condition, if the initial density contains vacuum at least at one point, then the global strong (or classical) solution must blow up as time goes to infinity. [Copyright &y& Elsevier]

Abstract: Consider classical solutions to the parabolic reaction diffusion equation whereis a nondegenerate elliptic operator, and the reaction term f converges to at a super-linear rate as . The first result in this paper is a parabolic Osserman–Keller type estimate. We give a sharp minimal growth condition on f, independent of L, in order that there exist a universal, a priori upper bound for all solutions to the above Cauchy problem—that is, in order that there exist a finite function on such that , for all solutions to the Cauchy problem. Assuming now in addition that , so that is a solution to the Cauchy problem, we show that under a similar growth condition, an intimate relationship exists between two seemingly disparate phenomena—namely, uniqueness for the Cauchy problem with initial data and the nonexistence of unbounded, stationary solutions to the corresponding elliptic problem. We also give a generic sufficient condition guaranteeing uniqueness for the Cauchy problem. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in . In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable. [Copyright &y& Elsevier]

Abstract: Practically every book on the Inverse Scattering Transform method for solving the Cauchy problem for KdV and other integrable systems refers to this method as nonlinear Fourier transform. If this is indeed so, the method should lead to a nonlinear analogue of the Fourier expansion formula . In this paper a special class of solutions of KdV whose role is similar to that of is discussed. The theory of these solutions, referred to here as harmonic breathers, is developed and it is shown that these solutions may be used to construct more general solutions of KdV similarly to how the functions are used to perform the same task in the theory of Fourier transform. A nonlinear superposition formula for general solutions of KdV similar to the Fourier expansion formula is conjectured. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the positive solution of the Cauchy problem for the equationand give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the Cauchy problem. Furthermore, the life span of solutions are also studied. [Copyright &y& Elsevier]

Abstract: The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the regularity, exponential stability of solutions and existence of universal attractor in for a nonlinear one-dimensional compressible Navier–Stokes equations describing a motion of heat-conductive viscous real gas in a bounded domain . Some new ideas and more delicated estimates are introduced to prove these results. [Copyright &y& Elsevier]

In this paper, we consider the Cauchy problem for the incompressible Navier–Stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field. For illustrative purposes, we first derive corresponding a priori estimates for certain parabolic systems. Because of the pressure term, the case of the Navier–Stokes equations is more difficult, however. [Copyright &y& Elsevier]