We study regularity properties for the solution of homogeneous boundary value problems for the anisotropic hyperbolic heat equation in the case of infinitely differentiable coefficients but irregular distributions as internal heat sources. | Turk J Math (2017) 41: 461 – 482 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Regularity of solutions of the anisotropic hyperbolic heat equation with nonregular heat sources and homogeneous boundary conditions ´ Juan Antonio LOPEZ MOLINA∗, Macarena TRUJILLO Department of Applied Mathematics, Polytechnic University of Valencia, Valencia, Spain • Received: Accepted/Published Online: • Final Version: Abstract: We study regularity properties for the solution of homogeneous boundary value problems for the anisotropic hyperbolic heat equation in the case of infinitely differentiable coefficients but irregular distributions as internal heat sources. Key words: Anisotropic hyperbolic heat equation, Sobolev spaces, integral transforms of vector valued distributions 1. Introduction and physical motivation The hyperbolic heat conduction equation is a fundamental tool in many modern industrial applications such as microelectronics and the processing of materials by irradiation with a laser beam of high intensity and very short application times (see [4, 6, 12, 13] for instance). Usually the mathematical formulation of these problems leads to the study of boundary value problems with data given by irregular distributions such as Heaviside’s function or Dirac’s δ distribution. Real industrial materials frequently are neither isotropic (see [16] for instance for some concrete examples) nor homogeneous. Assuming the density ρ and the specific heat c to be constant in order to avoid more complications, the hyperbolic heat equation in the open set Ω occupied by the body is (see [2]) ( ) 3 3 ∑ ∂ ∑ ∂T ∂T ∂2T − khj (x) (x, t) + ρ c (x, t) + τ (x, t) = ∂ xh j=1 ∂ xj ∂t ∂t2 h=1 ( ) ∂S = ρ S(x, t) + τ (x, t) , ∂t (1) where T (x, t) is the temperature in the point x at the instant t, (khj (x)) is the symmetric thermal conductivity tensor of the material, τ is .
