Remarks on Propagation of Singularities in Thermoelasticity

The propagation of high order weak singularities for the system of homogeneous thermoelasticity in one space variable is studied by using paralinearization and a new decoupling technique introduced by the author (Microlocal analysis in nonlinear thermoelasticity, to appear). For the linear system, one shows that the nonsmooth initial data for the parabolic part lead to singularities in the hyperbolic part of solutions, even when the initial data for that part are identically zero. Both the Cauchy problem and the problem inside of a domain for the semilinear system are considered. It is shown that the propagation of high order singularities is essentially dominated by the hyperbolic operator in the system of thermoelasticity.     

Propagation of singularities for Cauchy problems of semilinear thermoelastic systems with microtemperatures

The propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one space variable is studied. First, by using a diagonalization argument of phase space analysis, the coupled thermoelastic system with microtemperatures will be decoupled microlocally. Second, using a classical bootstrap argument, the property of finite propagation speed of singularities for the semilinear thermoelastic system is obtained. Finally, it is also shown that the microlocal weak singularities propagate along the null bicharacteristics of the hyperbolic operators of the coupled semilinear system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Propagation of Mild Singularities in Higher Dimensional Thermoelasticity

The propagation of mild singularities for the semilinear model of three-dimensional thermoelasticity is studied. It is shown that the propagation picture of such singularities of the solution to the semilinear model coincides with one of the solutions to the corresponding linear model. As a simple consequence of our method, a similar result for the full semilinear Cauchy problem of one-dimensional thermoelasticity is also presented.     

Edge Sobolev spaces and weakly hyperbolic equations

Summary. Edge Sobolev spaces are proposed as a main new tool for the investigation of weakly hyperbolic equations. The well-posedness of the linear and semilinear Cauchy problem in the class of these edge Sobolev spaces is proved. An application to the propagation of singularities for solutions to the semilinear problem is considered. Received: October 3, 2000 Published online: December 19, 2001     

Self-similar solutions of a semilinear parabolic equation with inverse-square potential

We investigate existence, nonexistence and asymptotical behaviour—both at the origin and at infinity—of radial self-similar solutions to a semilinear parabolic equation with inverse-square potential. These solutions are relevant to prove nonuniqueness of the Cauchy problem for the parabolic equation in certain Lebesgue spaces, generalizing the result proved by Haraux and Weissler [Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982) 167-189] for the case of vanishing potential.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A Probability Approach to the Method of Vanishing Viscosity

We clarify conditions under which solutions to the Cauchy problem for a general (fully nondiagonal) system of linear and nonlinear parabolic equations admit probability representations. Such representations are also used for constructing and studying solutions to the Cauchy problem for nonlinear hyperbolic systems. Bibliography: 26 titles.     

Grow-up rate of solutions for a supercritical semilinear diffusion equation

We study solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to a singular steady state from below as t→∞. We show that the grow-up rate of such solutions depends on the spatial decay rate of initial data.     

On the smoothness of solutions of an abstract coupled thermoelasticity problem

The Cauchy problem for a system of two operator-differential equations in Hilbert space that is a generalization of a number of linear coupled thermoelasticity problems is investigated. Results concerning the high smoothness of the solutions to these equations are proved.     

Rates of convergence to zero for a semilinear parabolic equation with a critical exponent

We study nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren. We establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate.     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.     

Sobolev multipliers,maximal functions and parabolic equations with a quadratic nonlinearity

We develop a general framework to describe global mild solutions to a Cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity. This class includes the Navier–Stokes equations, the subcritical dissipative quasi-geostrophic equation and the parabolic–elliptic Keller–Segel system.     

Formation of Singularities of Solutions for Cauchy Problem of Quasilinear Hyperbolic Systems with Dissipative Terms

ln this paper, for a class of 2 × 2 quasilinear hyperbolic systems, we get existence theorems of the global smooth solutions of its Cauchy problem, under a certain hypotheses. In addition, Tor two concrete quasilinear hyperbolic systems, we study the formation of the singularities of the C¹-solution to its Cauchy problem.     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.     

Existence and nonexistence of global solutions of the cauchy problem for semilinear hyperbolic equations with dissipation and with anisotropic elliptic part

We consider the Cauchy problem for a semilinear hyperbolic equation with anisotropic elliptic part and with dissipation. We prove existence and nonexistence theorems for global solutions.     

Very slow convergence rates in a semilinear parabolic equation

We consider the behavior of solutions to the Cauchy problem for a semilinear parabolic equation with supercritical nonlinearity. It is known that any slow algebraic rate appears as rate of convergence to regular steady states for some solutions. Now we show that this convergence takes place with arbitrarily slow rates which are slower than any algebraic rate, if the initial data are chosen suitably.     

The self-similar solutions of the one-dimensional semilinear Tricomi-type equations

In this article we investigate the issue of existence of global in time solutions of semilinear Tricomi-type equations. We give conditions that relate the nonlinearity, the speed of propagation, and the order of singularities of initial data. These conditions guarantee existence of global in time solutions. In particular, we prove existence of solutions invariant under dilation by solving the Cauchy problem with initial data which are homogeneous functions.     

H~1 SOLUTIONS FOR A CLASS OF SEMILINEAR PARABOLIC SYSTEMS

1IntroductionInthispapertweconsiderthefollowingsystemswithinitialvaluewhereu=(tLt,'',Wi),F=(FI,'.',Fn),t20,--co0,thereisasolutionu(x,f)of(1.1),suchthatIIullHI5C(T).C(T)isaconstantdependingboundlyon"o(z)andT.Ifb=0,then(1.l)isthetypicalselnilinearheatsystems,i.e.therearemugpapersdealingwiththesystems(1.3).Theinvestigationill…     

Slow convergence to zero for a parabolic equation with a supercritical nonlinearity

We study the behaviour of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t → ∞. We show that any algebraic decay rate slower than the self-similar one occurs for some initial data.     

Singularities of Solutions to Cauchy Problems for Semilinear Wave Equations in Two Space Dimensions

This paper concerns the Cauchy problem for semilinear wave equations with two space variables, of which the initial data have conormal singularities on finite curves intersecting at one point on the initial plane. It is proved that the solution is of conormal distribution type, and its singularities are contained in the union of the characteristic surfaces through these curves and the characteristic cone issuing from the intersection point.