Existence of solution of a forest fire spread model

We consider a wildfire spread model represented by the system (1). We use results from the theory of Hamilton-Jacobi equations to prove that there exists a classical solution of (1) for any (ϕ,t)∈R×(0,T) and some T>0 and satisfies particular initial conditions. We also use the method of characteristics to obtain the solution of (1) in a certain form.