On the heat equation involving the Dirac distribution as a coefficient

We consider the equation ( a H ( x + s t ) + b H ( − x − s t ) + δ ( x + s t ) ) u t ( x , t ) = u x x ( x , t ) , where ( x , t ) ∈ R × R + , a , b , s ∈ R are fixed constants, H is the Heaviside function, and δ 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.