An Operator Theoretical Approach to Nonlocal Differential Equations

An Operator Theoretical Approach to Nonlocal Differential Equations Joshua Lee Padgett Department of Mathematics and Statistics Texas Tech University Analysis Seminar November 27, 2017 Joshua Lee Padgett Texas Tech University November 27, 2017 1 / 34

Table of Contents 1 Motivation for the Proposed Approach 2 Background Material 3 Nonlocal Solution Operators 4 Nonlocal Kawarada Problem 5 Future Work Joshua Lee Padgett Texas Tech University November 27, 2017 2 / 34

Table of Contents Motivation for the Proposed Approach 1 Motivation for the Proposed Approach 2 Background Material 3 Nonlocal Solution Operators 4 Nonlocal Kawarada Problem 5 Future Work Joshua Lee Padgett Texas Tech University November 27, 2017 2 / 34

Motivation Motivation for the Proposed Approach Let Ω be a bounded open domain in R d with smooth boundary Ω. Then consider the following semilinear problem: t u = u + f (u) (x, t) Ω (0, T ) u = 0 (x, t) Ω (0, T ) u = u 0 (x, t) Ω {t = 0} where 1. u 0 is positive and appropriately smooth 2. f is locally Lipschitz on [0, c) = f (u) f (v) L f (c) u v, 0 u, v < c 3. f : [0, c) R + is a monotonically increasing convex function such that lim f (u) = + and u c c 0 f (u) du = + (1.1) Joshua Lee Padgett Texas Tech University November 27, 2017 3 / 34

Motivation Motivation for the Proposed Approach (1.1) is known as the Kawarada problem and is a model for numerous phenomenon such as solid-fuel combustion and MEMS devices. = (1.1) is of mathematical interest due to the unique quenching feature of it solution under certain conditions. Definition A solution u of (1.1) is said to quench in finite time if there exists a T < such that max u c, as t T (1.2) x Ω If (1.2) holds for T =, then u is said to quench in infinite time. A critical question is when and where does (1.2) occur? Joshua Lee Padgett Texas Tech University November 27, 2017 4 / 34

Motivation Motivation for the Proposed Approach Ultimately, I am concerned with the numerical approximation of the solution to (1.1) that is valid regardless of whether the solution quenches. There are numerous possibilities for solving (1.1) numerically. = Roughly speaking, there are two basic options: 1. Discretize everything in sight" and then solve the resulting difference equation (exactly or by other means). 2. Solve the original problem exactly" and then approximate the resulting solution. = This is often carried out by a method known as semi-discretization. The second option is MY preferred approach and is the motivation for my current explorations. Joshua Lee Padgett Texas Tech University November 27, 2017 5 / 34

Motivation Motivation for the Proposed Approach In order to develop fully robust numerical methods via the second option (from the previous slide), we need to understand the solution to (1.1) in terms of certain operators. We recast the problem into some appropriate Banach space X. u (t) = Au(t) + f (u(t)) 0 < t < T (1.3) u(0) = u 0 Then the (mild) solution is (formally) given by t u(t) = E(t)u 0 + E(t s)f (u(s)) ds, 0 t < T, (1.4) 0 where E(t) is the semigroup generated by A. Joshua Lee Padgett Texas Tech University November 27, 2017 6 / 34

Motivation Motivation for the Proposed Approach We can use (1.4) to develop numerical solutions to the problem (1.1). = This involves several aspects. 1. Approximating A appropriately. 2. Approximating E(t) appropriately. 3. Approximating the integral portion appropriately. This method works nicely because the semigroup E(t) has many nice properties that we are able to take advantage of! = Why would we employ such a method when there are numerous other possibilities that require less" work? STABILITY and EXPLICITNESS! Joshua Lee Padgett Texas Tech University November 27, 2017 7 / 34

Table of Contents Background Material 1 Motivation for the Proposed Approach 2 Background Material 3 Nonlocal Solution Operators 4 Nonlocal Kawarada Problem 5 Future Work Joshua Lee Padgett Texas Tech University November 27, 2017 8 / 34

Background Material Nonlocal Kawarada Problem We now generalize our model significantly. Banach space. D(A) is the domain of A σ(a) is the spectrum of A ρ(a) := C σ(a) is the resolvent set of A Let (X, ) be a (real) The family R(z; A) = (zi A) 1, z ρ(a) of bounded linear operators is the resolvent of A L (Y, Z ) is the space of all bounded linear operators between two normed spaces Y and Z with the operator norm L (Y,Z ) = We abbreviate this notation to L (Y ) when Y = Z, and write T L (X) as T for every T L (X) when it has no loss of clarity Joshua Lee Padgett Texas Tech University November 27, 2017 9 / 34

Background Material Nonlocal Kawarada Problem We now consider { α t u(t) = Au(t) + λf (u(t)) u(0) = u 0 0 < t < T (2.5) where 1. A is an almost sectorial operator 2. α t is the α order Caputo fractional derivative with α (0, 1) 3. λ > 0 = This parameter λ results from a rescaling of the domain Ω; i.e., λ = λ(ω, α, A). Can we develop a solution similar to (1.4) for (2.5)?? Joshua Lee Padgett Texas Tech University November 27, 2017 10 / 34

Background Material Almost Sectorial Operators Definition Let 1 < γ < 0 and 0 < ω < π/2. By Θ γ ω(x) we denote the family of all linear closed operators A : D(A) X X which satisfy 1. σ(a) S ω := {z C\{0} : arg z ω} {0} and 2. for every ω < µ < π there exists a constant C µ such that R(z; A) C µ z γ, z C\S µ. (2.6) A linear operator A will be called an almost sectorial operator on X if A Θ γ ω. Joshua Lee Padgett Texas Tech University November 27, 2017 11 / 34

Background Material Almost Sectorial Operators We outline a few details regarding the previous definition. It is worth noting that the case when γ = 1 results in A being a sectorial operator. If A Θ γ ω(x), then the definition implies that 0 ρ(a). We say that the estimate (2.6) is deficient" since γ > 1. If A Θ γ ω(x), then A generates a semigroup E(t) with a singular behavior at t = 0 in a sense, called a semigroup of growth 1 + γ. = The semigroup E(t) is analytic in an open sector of the complex plane C, but the strong continuity fails at t = 0 for data which are not sufficiently smooth. Joshua Lee Padgett Texas Tech University November 27, 2017 12 / 34

Background Material Almost Sectorial Operators It is well known that many elliptic differential operators considered on the space of continuous functions or the Lebesgue spaces belong to the class of sectorial operators. = Therefore, many PDEs with elliptic operators can be transformed into evolution equations with sectorial operators in a Banach space. However, when considering elliptic differential operators on spaces of regular functions, such as the space of Hölder continuous functions, the operators fail to be sectorial. = The resolvent estimate fails to hold. = This leads to the consideration of almost sectorial operators. Joshua Lee Padgett Texas Tech University November 27, 2017 13 / 34

Background Material An Example of an Almost Sectorial Operators Assume that Ω is a bounded domain in R d (d 1) with boundary Ω of class C 4. Let X = C l (Ω), l (0, 1), denote the Banach space of Hölder continuous functions with usual norm l. Consider the elliptic differential operator A : D(A) X X in the form Au := u(x), u D(A) := {u C 2+l (Ω) : u Ω = 0}. a. A is not densely defined in C l (Ω) b. There exists ν, ɛ > 0 such that σ(a + ν) S π/2 ɛ := {λ C\{0} : arg λ π/2 ɛ} {0} R(λ; A + ν) L (C l (Ω)) C λ l/2 1, λ C\S π/2 ɛ c. The exponent l/2 1 ( 1, 0) is sharp (In particular, A + ν is not sectorial) Joshua Lee Padgett Texas Tech University November 27, 2017 14 / 34

Caputo Derivative Background Material In the following, we let I = (0, T ) and Γ( ) be Euler s gamma function. Further, for α > 0 we define the following function g α (t) = { t α 1 /Γ(α), t > 0, 0, t 0, (2.7) with g 0 (t) 0. Definition Let u L 1 (I) and α 0. The Riemann-Liouville fractional integral of order α of u is defined as Jt α u(t) = (g α u)(t) = where Jt 0 u(t) = u(t). t 0 g α (t s)u(s) ds, t > 0, Joshua Lee Padgett Texas Tech University November 27, 2017 15 / 34

Caputo Derivative Background Material Definition Let u C m 1 (I) and g m α u W m,1 (I) where m N and 0 m 1 < α m. The Riemann-Liouville fractional derivative of order α of u is defined by RD α t u(t) := D m t (g m α u)(t) = D m t J m α t u(t), t > 0, where D m t := d m /dt m. Joshua Lee Padgett Texas Tech University November 27, 2017 16 / 34

Caputo Derivative Definition Background Material Let u C m 1 (I) and g m α u W m,1 (I) where m N and 0 m 1 < α m. The regularized Caputo fractional derivative of order α of u is defined as α t u(t) = D m t J m α t ( u(t) m 1 i=1 f (i) (0)g i+1 (t) ), t > 0, (2.8) where Dt m := d m /dt m. If u is continuously differentiable with respect to t, then t α d m /dt m as α m. We note that for α (0, 1), if u is smooth enough, the Caputo fractional derivative can be written as α t u(t) = 1 Γ(1 α) t 0 (t s) α u (s) ds. Joshua Lee Padgett Texas Tech University November 27, 2017 17 / 34

Background Material Mittag-Leffler Functions Finally, we introduce an important class of functions associated with fractional calculus, known as the Mittag-Leffler functions. Definition Let α, β C with Re(α), Re(β) > 0. Then we may define the generalized Mittag-Leffler function to be E α,β (z) := n=0 z n Γ(β + αn) = 1 λ α β e λ 2πi γ λ α z dλ where γ is a contour which starts at and encircles the disc λ z 1/α counterclockwise. Joshua Lee Padgett Texas Tech University November 27, 2017 18 / 34

Background Material Mittag-Leffler Functions For 0 < α < 1, β > 0, we have as z { α E α,β (z) = 1 z (1 β)/α exp(z 1/α ) + ε α,β (z), arg z απ/2, ε α,β (z), arg z < (1 α/2)π, where We also have N 1 z n ( ε α,β (z) = Γ(β αn) + O z N), as z. n=1 α t E α,1 (ωt α ) = ωe α,1 (ωt α ), ( ) J 1 α t t α 1 E α,α (ωt α ) = E α,1 (ωt α ). That is, the function E α,1 is the eigenfunction corresponding to the Caputo fractional derivative. Further, we have E 1,1 (z) = e z. Joshua Lee Padgett Texas Tech University November 27, 2017 19 / 34

Background Material Mittag-Leffler Functions Consider also the Wright function, for z C with 0 < α < 1, given by Ψ α (z) := n=0 ( z) n n!γ(1 (n + 1)α) = 1 π n=1 ( z) n Γ(nα) sin(nπα), (n 1)! Lemma (Wang, et. al. (2012)) Let 0 < α < 1. For 1 < r <, λ > 0, the following results hold. i. Ψ α (t) 0, t > 0; ii. 0 αt α 1 Ψ α (t α )e λt dt = e λα ; iii. 0 Ψ α(t)t r dt = Γ(1 + r)/γ(1 + αr); iv. 0 Ψ α(t)e zt dt = E α,1 ( z), z C; v. 0 αtψ α(t)e zt dt = E α,α ( z), z C. Joshua Lee Padgett Texas Tech University November 27, 2017 20 / 34

Table of Contents Nonlocal Solution Operators 1 Motivation for the Proposed Approach 2 Background Material 3 Nonlocal Solution Operators 4 Nonlocal Kawarada Problem 5 Future Work Joshua Lee Padgett Texas Tech University November 27, 2017 21 / 34

Solution Operators Nonlocal Solution Operators We now introduce the following solution operators E(t) := e tz (A) = 1 e tz R(z; A) dz, (3.9) 2πi Γ θ S α (t) := E α,1 ( zt α )(A) = 1 E α,1 ( zt α )R(z; A) dz, (3.10) 2πi Γ θ P α (t) := E α,α ( zt α )(A) = 1 E α,α ( zt α )R(z; A) dz, (3.11) 2πi Γ θ for t S 0 π/2 ω and z C\(, 0], where the path Γ θ := {R + e iθ } {R + e iθ } (0 < θ < π) is oriented so that S 0 θ lies to the left of Γ θ. = In the following slides we present some known results regarding these operators for completeness. Joshua Lee Padgett Texas Tech University November 27, 2017 22 / 34

Nonlocal Solution Operators Properties of the Solution Operators Lemma (Wang, et. al. (2012)) Let A Θ γ ω(x) with 1 < γ < 0 and 0 < ω < π/2. Then i. E(t) is analytic in S 0 π/2 ω and d n /dt n E(t) = ( A) n E(t) (t S 0 π/2 ω ) ii. E(s + t) = E(s)E(t) for all s, t S 0 π/2 ω iii. C 0 = C 0 (γ) > 0 such that E(t) C 0 t γ 1 (t > 0) iv. The range R(E(t)) of E(t), t S 0 π/2 ω, is contained in D(A ) v. If β > 1 + γ, then D(A β ) Σ E = {x X : lim t 0 + E(t)x = x} vi. λ C with Re(λ) > 0, one has R(λ; A) = 0 e λt E(t) dt = Note that condition ii. does not hold for t = 0 or s = 0. Joshua Lee Padgett Texas Tech University November 27, 2017 23 / 34

Nonlocal Solution Operators Properties of the Solution Operators While the operators S α (t) and P α (t) are not semigroups, they behave nicely" and have some desirable properties that are similar to semigroups. = S α (t), P α (t) E(t) as α 1 Lemma Let A Θ γ ω with 1 < γ < 0 and 0 < ω < π/2. Then for each fixed t S 0 π/2 ω, S α(t) and P α (t) are linear bounded operators on X. Moreover, there exist constants C s = C(α, γ) > 0 and C p = C p (α, γ) > 0 such that for all t > 0, S α (t) C s t α(1+γ), P α (t) C p t α(1+γ) Joshua Lee Padgett Texas Tech University November 27, 2017 24 / 34

Nonlocal Solution Operators Properties of the Solution Operators Lemma (Wang, et. al. (2012)) Let A Θ γ ω with 1 < γ < 0 and 0 < ω < π/2. Then i. For t > 0, S α (t) and P α (t) are continuous in the uniform operator topology (actually uniformly) ii. For t S 0 π/2 ω and x D(A), (S α(t) I)x = t 0 sα 1 AP α (s)x ds iii. For all x D(A) and t > 0, α t S α (t)x = AS α (t)x iv. For all t > 0, S α (t) = J α t (t α 1 P α (t)) v. Let β > 1 + γ, then for all x D(A β ), lim t 0 + S α (t)x = x = There are many other useful/necessary properties, but we will hold off since we will not be proving anything. Joshua Lee Padgett Texas Tech University November 27, 2017 25 / 34

Table of Contents Nonlocal Kawarada Problem 1 Motivation for the Proposed Approach 2 Background Material 3 Nonlocal Solution Operators 4 Nonlocal Kawarada Problem 5 Future Work Joshua Lee Padgett Texas Tech University November 27, 2017 26 / 34

Nonlocal Kawarada Problem Nonlocal Kawarada Problem We again consider (2.5): { α t u(t) = Au(t) + λf (u(t)) 0 < t < T u(0) = u 0 1. The problem is considered on a Banach space X that satisfies an interior cone condition (this allows for a notion of positivity). 2. A Θ γ ω(x) with 1 < γ < 0 and 0 < ω < π/2. 3. f is now defined on some ball of radius c contained in X (and maps into X). 4. f is still monotonically increasing and singular on the boundary of its domain. Joshua Lee Padgett Texas Tech University November 27, 2017 27 / 34

Nonlocal Kawarada Problem Nonlocal Kawarada Problem What do we mean by a solution to (2.5)? Definition A function u X is a mild solution to (2.5) if u < c and for any t (0, T ] u(t) = S α (t)u 0 + t 0 (t s) α 1 P α (t s)f (u(s)) ds (4.12) = This notion of a solution is very similar to (1.4)! = This form is derived via Laplace transform methods. Approximating the solution to (2.5) amounts to approximating the operators S α (t) and P α (t)! Joshua Lee Padgett Texas Tech University November 27, 2017 28 / 34

Nonlocal Kawarada Problem Local Existence and Uniqueness Theorem Assume that A Θ γ ω(x) with 1 < γ < 0 and 0 < ω < π/2 and that u 0 D(A β ) for β > 1 + γ. Then there exists a T = T (λ, α) > 0 such that a unique mild solution to (2.5) exists on [0, T ]. = This mirrors the classical" result for such problems. = We are actually able to derive an upper bound on the time of existence through the Cauchy problem { α t v(t) = t α(1+γ) Φ(Ct α(1+γ) v(t)) 0 < t < T v(0) = u 0 where Φ(z) := sup u z f (u). Joshua Lee Padgett Texas Tech University November 27, 2017 29 / 34

Nonlocal Kawarada Problem Positivity and Monotonicity Theorem If X satisfies the interior cone condition and u 0 is positive then u(t) is positive on its interval of existence. Theorem If X satisfies the interior cone condition and Au 0 + f (u 0 ) > 0, then the sequence {u(t)} t (0,T ] is monotonically increasing on its interval of existence. = This monotonicity can be viewed as u(t) u(s) > 0 for each 0 < s < t or as (Lu(t) Lu(s), u(t) u(s)) > 0 for all 0 < s < t, where L is the solution operator." Joshua Lee Padgett Texas Tech University November 27, 2017 30 / 34

Nonlocal Kawarada Problem Why Do We Care About This Approach? With the numerous numerical methods available, why do we care about this new approach? = These fractional derivative" operators are nonlocal! This means that standard numerical methods can be quite costly as the resulting discretizations will result in dense matrices. = This operator approach removes the nonlocal features of the time portion of the problem by approximating the Mittag-Leffler operators directly. Joshua Lee Padgett Texas Tech University November 27, 2017 31 / 34

Future Work Table of Contents 1 Motivation for the Proposed Approach 2 Background Material 3 Nonlocal Solution Operators 4 Nonlocal Kawarada Problem 5 Future Work Joshua Lee Padgett Texas Tech University November 27, 2017 32 / 34

Future Work Future Work My primary concerns with this type of approach fall into two categories: 1. The development of splitting" or composition" methods for Mittag-Leffler operators. = This would mirror the classical splitting ideas for standard semigroups. 2. The development of a Magnus-type solution operator for abstract fractional Cauchy problems. = Can we write the solution to (1.1) as for some t (0, T )? u(t) = E α,1 ( zt α )(Ω(t, u(t))u 0 Joshua Lee Padgett Texas Tech University November 27, 2017 33 / 34

THANK YOU! Questions? Joshua Lee Padgett Texas Tech University November 27, 2017 34 / 34