Global existence and blowup in infinite time for a fourth order wave equation with damping and logarithmic strain terms

<p style='text-indent:20px;'>We consider the well-posedness of solution of the initial boundary value problem to the fourth order wave equation with the strong and weak damping terms, and the logarithmic strain term, which was introduced to describe many complex physical processes. The local solution is obtained with the help of the Galerkin method and the contraction mapping principle. The global solution and the blowup solution in infinite time under sub-critical initial energy are also established, and then these results are extended in parallel to the critical initial energy. Finally, the infinite time blowup of solution is proved at the arbitrary positive initial energy.</p>

DYNAMICS OF UNSTABLE SOLUTIONS FOR THE WAVE EQUATION WITH SOURCES

Two new accurate solutions of the wave equation with sources are obtained. The dynamics of unstable states described by these solutions is studied. Analytical forms are given for the partial derivatives of the required function with respect to the spatial coordinate and time on the plane of independent variables “the required function – time”. This structure of the solution allows us to consider nonstationary analogs of self-similar kinks describing the transition between two equilibrium states of the “medium – source” system. For the classical wave equation, a nonlinear rheonomic source is used, the behavior of which affects the properties of the relaxing kink. The conditions under which the speed of movement of the formed self-similar transfer wave is subsonic or supersonic are determined. An important role of the velocity of the inflection point of a nonselfsimilar kink has been analyzed; the threshold value of the velocity is calculated, which separates the subsonic and supersonic regimes. An unstable version of the presented solution gives a strong discontinuity of the required function with an unlimited increase in time. The stopping of the kink inflection point is an indicator of a strong rupture. An estimate of the value of the moment in time preceding the beginning of the return motion of the inflection point is indicated. A solution to a spatially nonlocal fourth-order wave equation with two additively entering sources is given. One source depends on the desired function in a linear homogeneous way; the second one depends on the modulus of the gradient of the desired function the same way. The solution is an analog of an overthrow wave in an interval with non-stationary boundaries. At each finite moment of time this solution is continuous, and for an infinite time there is a loss of smoothness of the solution, we have the so-called “slow explosion”. In the unstable solution, the isolines of the sought-for function on the concave section (the lower part of the kink) move towards the convex section, which is adjacent to the upper boundary of the kink. In the stable version, the kink degenerates into a homogeneous state. It has been analyzed that for a nonselfsimilar process, the inversion of the sign of the gradient source gives an inversion of the stability conditions for the kink and antikink. An unstable kink/ antikink corresponds to a gradient sink/source.

Fourth-order wave equation in Bhabha–Madhavarao spin-3 2 theory

Within the framework of the Bhabha–Madhavarao formalism, a consistent approach to the derivation of a system of the fourth-order wave equations for the description of a spin-[Formula: see text] particle is suggested. For this purpose an additional algebraic object, the so-called [Formula: see text]-commutator ([Formula: see text] is a primitive fourth root of unity) and a new set of matrices [Formula: see text], instead of the original matrices [Formula: see text] of the Bhabha–Madhavarao algebra, are introduced. It is shown that in terms of the [Formula: see text] matrices we have succeeded in reducing a procedure of the construction of fourth root of the fourth-order wave operator to a few simple algebraic transformations and to some operation of the passage to the limit [Formula: see text], where [Formula: see text] is some (complex) deformation parameter entering into the definition of the [Formula: see text]-matrices. In addition, a set of the matrices [Formula: see text] and [Formula: see text] possessing the properties of projectors is introduced. These operators project the matrices [Formula: see text] onto the spins 1/2- and 3/2-sectors in the theory under consideration. A corresponding generalization of the obtained results to the case of the interaction with an external electromagnetic field introduced through the minimal coupling scheme is carried out. The application to the problem of construction of the path integral representation in para-superspace for the propagator of a massive spin-[Formula: see text] particle in a background gauge field within the Bhabha–Madhavarao approach is discussed.