We study the Dirichlet problem for the parabolic equation u t = u m m > 0, in a bounded, non-cylindrical and non-smooth domain ⊂ N+1 N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower (or left-upper) semicontinuity at the points of the lateral boundary manifold and show that the upper (or lower) Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution. The Hölder exponent 1 2 is critical as in the classical theory of the one-dimensional heat equation u t = u xx .

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of L 2 -norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.Key words: Inverse Stefan problem, optimal control, second order parabolic PDE, Sobolev spaces, energy estimate, embedding theorems, traces of Sobolev functions, method of lines, discrete optimal control problem, convergence in functional, convergence in control.AMS subject classifications: 35R30, 35R35, 35K20, 35Q93, 65M32, 65N21. Proof of Theorem 1.2We split the remainder of the proof into three lemmas. Lemma 3.5 Let J * (±ǫ) = inf V R±ǫ J (v), ǫ > 0. Then lim ǫ→0 J * (ǫ) = J * = lim ǫ→0 J * (−ǫ) (3.55)Proof: Note that for 0 < ǫ 1 < ǫ 2 we have

Abstract. We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtrationThe interface may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In all cases, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. The methods of the proof are based on nonlinear scaling laws, and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. InrtroductionWe consider the Cauchy problem(CP) for the nonlinear degenerate parabolic equationwhere p > 2, b ∈ R, β > 0, 0 < T ≤ +∞, and u 0 is nonnegative and continuous. We assume that b > 0 if β < 1, and b is arbitrary if β ≥ 1 (see Remark 1.1). Equation (1.1) arises in many applications, such as the filtration of non-Newtonian fluids in porous media ([8] or nonlinear heat conduction ([9]) in the presence of the reaction term expressing additional release (b > 0) or absorption (b < 0) of energy. The goal of this paper is to analyze the behavior of interfaces separating the regions where u = 0 and where u > 0. We present full classification of the short-time evolution of interfaces and local structure of solutions near the interface. Due to invariance of (1.1) with respect to translation, without loss of generality, we will investigate the case when η(0) = 0, where η(t) = sup {x : u(x, t) > 0}. and precisely, we are interested in the short-time behavior of the interface function η(t) and local solution near the interface. We shall assume thatThe direction of the movement of the interface and its asymptotics is an outcome of the competition between the diffusion and reaction terms and depends on the parameters p, b, β, C, and α. Since the main results are local in nature, without loss of generality we may suppose that u 0 either is bounded or satisfies some 1 arXiv:1605.07279v1 [math.AP]

We consider the Cauchy Dirichlet and Dirichlet problems for the nonlinear parabolic equation