On the heat equation involving the Dirac δ distribution as a coefficient

We consider the equation (aH(x + st) + bH(-x - st) + delta(x + st)u(t)(x, t) = u(xx)(x, t), where (x, t) is an element of R X R+, a, b, s is an element of R are fixed constants, H is the Heaviside function, and delta is the Dirac distribution. We augment the equation with appropriate initial and boundary data. We give a physical model justifying such an equation, and introduce a new solution concept with the help of a distribution space defined on discontinuous test functions. We prove the existence and uniqueness of a solution in the framework of the proposed solution concept. (C) 2009 Elsevier Ltd. All rights reserved.