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

We consider the equation (aH(x+st)+bH(−x−st)+δ(x+st))ut(x,t)=uxx(x,t),(aH(x+st)+bH(−x−st)+δ(x+st))ut(x,t)=uxx(x,t), where (x,t)∈R×R+, a,b,s∈R are fixed constants, HH is the Heaviside function, and δδ 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.