Abstract We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is with respect to both the unknown and its gradient, homogeneous, depends only on direction of gradient. For such equation, we obtain existence uniqueness entropy solutions Dirichlet problem, homogeneous Neumann Cauchy problem. Qualitative properties solutions, as finite speed propa...

We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator A and Caputo fractional derivative order α∈(0,2) time. The previously known representation mild solution to such does not have conventional variation-of-constants like form, with propagator derived from associated homogeneous problem. Instead, it relies on existence two propag...

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation ut + (−△) u = F (u) for initial data in the Lebesgue space L(R) with r ≥ rd , nb/(2α− d) or the homogeneous Besov space Ḃ p,∞(R ) with σ = (2α − d)/b − n/p and 1 ≤ p ≤ ∞, where α > 0, F (u) = f(u) or Q(D)f(u) with Q(D) being a homogeneous pseudo-differential operator of order d ∈ [0, 2α) and f(u) is a ...

This paper presents several characterizations of a local α-times integrated C-semigroup {T(t); 0 ≤ t < τ} by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem.We also discuss properties concerning the nondegeneracy of T(·), the injectivity of C, the closability of subgenerators, the commutativity of T(·), and extension of solutions of the as...