Exact Solutions of the Equation of a Nonlinear Conductor Model
- PDF / 325,933 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 22 Downloads / 289 Views
IAL DIFFERENTIAL EQUATIONS
Exact Solutions of the Equation of a Nonlinear Conductor Model A. I. Aristov1∗ 1
Lomonosov Moscow State University, Moscow, 119991 Russia e-mail: ∗ [email protected]
Received June 23, 2020; revised June 23, 2020; accepted June 26, 2020
Abstract—We consider a nonlinear Sobolev equation of the third order describing the electric field potential in a conductor. Six classes of exact solutions are constructed, and their qualitative behavior is analyzed. DOI: 10.1134/S0012266120090013
INTRODUCTION This paper deals with constructing and studying exact solutions of the equation ∂ ∆u = div (∆u∇u), ∂t
(1)
where the function u = u(x, t) depends on the spatial variables x = (x1 , x2 , x3 ) ∈ R3 and the time variable t > 0. The operators 4, ∇, and div are assumed to act only with respect to the spatial variables xk , k = 1, 2, 3. Equation (1) was studied in [1], where this equation was derived based on a model in which u (taken with the opposite sign) has the meaning of the electric field potential in a conductor occupying a bounded domain in R3 . In addition, the paper [1] used the trial function method to obtain sufficient conditions for the time-global unsolvability of the classical initial–boundary value problems for Eq. (1) and establish estimates of the solution lifespan. The blowup of solutions in finite time (the lack of time-global solvability) has the physical meaning of a breakdown in the conductor. Note that there is an extensive body of research into the qualitative theory of Sobolev equations examining the existence and uniqueness of solutions as well as their blowup and asymptotics (see, e.g., [2–4]), but Sobolev type equations rarely occur in the literature on exact solutions of partial differential equations (see, e.g., [5–7]). In the present paper, we construct several classes of exact solutions of Eq. (1) expressed via elementary and special functions and analyze the qualitative behavior of these solutions. Theorem. There exist exact solutions of Eq. (1) that are expressed via elementary functions and can exhibit one of the following types of qualitative behavior: (a) They tend to infinity as t tends to some finite value. (b) They are bounded on each time interval (but not globally). (c) They are bounded globally in time. The validity of this assertion follows from the considerations to follow, where we construct appropriate examples for each of the types (a)–(c). In the sequel, by c, c1 , c2 , . . . we denote, generally speaking, arbitrary real constants; by α, an arbitrary constant nonzero three-dimensional vector (unless specified otherwise). We assume that the parameters and variables are such that the transformations performed are correct. 1. RUNNING WAVE METHOD We seek solutions of Eq. (1) in the form u(x, t) = v(ξ), where ξ = hα, xi + t, α is an arbitrary constant nonzero three-dimensional vector, and h · , · i is the inner product on R3 . Substituting this expression into the equation, we bring it to the form 0
hα, αiv 000 (ξ) = hα, αi2 (v 0 (ξ)v 00 (ξ)) . 1113
1114
Data Loading...
