Nonuniqeness in a free boundary problem from combustion Arshak - - PowerPoint PPT Presentation

Nonuniqeness in a free boundary problem from combustion

Arshak Petrosyan

joint with Aaron Yip

Department of Mathematics Purdue University West Lafayette, IN 47907, USA

SIAM PD07

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Given u ∈ C(n), u ≥ 

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Given u ∈ C(n), u ≥  ▸ Find u ∶ n × [, ∞) → , u ≥ :

∆u − ∂tu =  in {u > } ∣∇u∣ = 

• n ∂{u > }

u(⋅, ) = u

• n n

(P)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Given u ∈ C(n), u ≥  ▸ Find u ∶ n × [, ∞) → , u ≥ :

∆u − ∂tu =  in {u > } ∣∇u∣ = 

• n ∂{u > }

u(⋅, ) = u

• n n

(P)

u >  ∆u − ∂tu =  u ≡  ∣∇u∣ = 

✁ ✁ ☛

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Given u ∈ C(n), u ≥  ▸ Find u ∶ n × [, ∞) → , u ≥ :

∆u − ∂tu =  in {u > } ∣∇u∣ = 

• n ∂{u > }

u(⋅, ) = u

• n n

(P)

unburnt zone flame front

✁ ✁ ☛

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Appears as the limit as ε → + of the singular perturbation problem

∆uε − ∂tuε = βε(uε) in n × (, ∞) uε(⋅, ) = uε



• n n,

(Pε) where uε

 ≈ u and βε ∈ Lip() satisfies

βε ≥ , supp βε = [, ε], ∫

ε  βε(s)ds = 



• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Appears as the limit as ε → + of the singular perturbation problem

∆uε − ∂tuε = βε(uε) in n × (, ∞) uε(⋅, ) = uε



• n n,

(Pε) where uε

 ≈ u and βε ∈ Lip() satisfies

βε ≥ , supp βε = [, ε], ∫

ε  βε(s)ds = 



▸ Describes the evolution of equidiffusional flames in the limit of high

activation energy.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

A free boundary problem from combustion

One-phase parabolic free boundary problem with fixed gradient condition

▸ Appears as the limit as ε → + of the singular perturbation problem

∆uε − ∂tuε = βε(uε) in n × (, ∞) uε(⋅, ) = uε



• n n,

(Pε) where uε

 ≈ u and βε ∈ Lip() satisfies

βε ≥ , supp βε = [, ε], ∫

ε  βε(s)ds = 



▸ Describes the evolution of equidiffusional flames in the limit of high

activation energy.

▸ Goes back to Zeldovich and Frank-Kamenetski in 1930’s

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Classical solutions

How to understand the condition ∣∇u∣ =  on the free boundary ∂{u > }?

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Classical solutions

How to understand the condition ∣∇u∣ =  on the free boundary ∂{u > }?

▸ Supersolutions:

limsup

{u>}∋(x,t)→(x,t)

∣∇u(x, t)∣ ≤  for any (x, t) ∈ ∂{u > }

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Classical solutions

How to understand the condition ∣∇u∣ =  on the free boundary ∂{u > }?

▸ Supersolutions:

limsup

{u>}∋(x,t)→(x,t)

∣∇u(x, t)∣ ≤  for any (x, t) ∈ ∂{u > }

▸ Subsolutions:

liminf

{u>}∋(x,t)→(x,t)∣∇u(x, t)∣ ≥ 

for any (x, t) ∈ ∂{u > }.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Classical solutions

How to understand the condition ∣∇u∣ =  on the free boundary ∂{u > }?

▸ Supersolutions:

limsup

{u>}∋(x,t)→(x,t)

∣∇u(x, t)∣ ≤  for any (x, t) ∈ ∂{u > }

▸ Subsolutions:

liminf

{u>}∋(x,t)→(x,t)∣∇u(x, t)∣ ≥ 

for any (x, t) ∈ ∂{u > }.

▸ Short time existence and uniqueness, provided ∂{u > } and u are

smooth [Baconneau-Lunardi 04]

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Classical solutions

How to understand the condition ∣∇u∣ =  on the free boundary ∂{u > }?

▸ Supersolutions:

limsup

{u>}∋(x,t)→(x,t)

∣∇u(x, t)∣ ≤  for any (x, t) ∈ ∂{u > }

▸ Subsolutions:

liminf

{u>}∋(x,t)→(x,t)∣∇u(x, t)∣ ≥ 

for any (x, t) ∈ ∂{u > }.

▸ Short time existence and uniqueness, provided ∂{u > } and u are

smooth [Baconneau-Lunardi 04]

▸ Generally classical solutions will develop singularities afer some time

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Classical solutions

How to understand the condition ∣∇u∣ =  on the free boundary ∂{u > }?

▸ Supersolutions:

limsup

{u>}∋(x,t)→(x,t)

∣∇u(x, t)∣ ≤  for any (x, t) ∈ ∂{u > }

▸ Subsolutions:

liminf

{u>}∋(x,t)→(x,t)∣∇u(x, t)∣ ≥ 

for any (x, t) ∈ ∂{u > }.

▸ Short time existence and uniqueness, provided ∂{u > } and u are

smooth [Baconneau-Lunardi 04]

▸ Generally classical solutions will develop singularities afer some time ▸ If u is radially symmetric, the solutions will stay classical until its

extinction, i.e when u becomes identically  [Galaktionov-Hulshof-V

azquez 97]

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Limit solutions

Most natural notion of solution of (P) reflecting its origin.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Limit solutions

Most natural notion of solution of (P) reflecting its origin.

▸ Let uε  → u in the sense

∥uε

 − u∥L∞(n) → ,

suppuε

 → suppu,

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Limit solutions

Most natural notion of solution of (P) reflecting its origin.

▸ Let uε  → u in the sense

∥uε

 − u∥L∞(n) → ,

suppuε

 → suppu, ▸ Solve the approximating problem (Pε)

∆uε − ∂tuε = βε(uε) in n × (, ∞), uε(⋅, ) = uε

.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Limit solutions

Most natural notion of solution of (P) reflecting its origin.

▸ Let uε  → u in the sense

∥uε

 − u∥L∞(n) → ,

suppuε

 → suppu, ▸ Solve the approximating problem (Pε)

∆uε − ∂tuε = βε(uε) in n × (, ∞), uε(⋅, ) = uε

. ▸ Te family {uε}ε> will be uniformly bounded in C, x ∩ C,/ t

norm on any K ⋐ n × (, ∞) [Caffarelli-V

azquez 95]

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Limit solutions

Most natural notion of solution of (P) reflecting its origin.

▸ Let uε  → u in the sense

∥uε

 − u∥L∞(n) → ,

suppuε

 → suppu, ▸ Solve the approximating problem (Pε)

∆uε − ∂tuε = βε(uε) in n × (, ∞), uε(⋅, ) = uε

. ▸ Te family {uε}ε> will be uniformly bounded in C, x ∩ C,/ t

norm on any K ⋐ n × (, ∞) [Caffarelli-V

azquez 95]

▸ Hence, for a subsequence ε = ε j → +,

uε j → u locally uniformly in n × (, ∞)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Limit solutions

Most natural notion of solution of (P) reflecting its origin.

▸ Let uε  → u in the sense

∥uε

 − u∥L∞(n) → ,

suppuε

 → suppu, ▸ Solve the approximating problem (Pε)

∆uε − ∂tuε = βε(uε) in n × (, ∞), uε(⋅, ) = uε

. ▸ Te family {uε}ε> will be uniformly bounded in C, x ∩ C,/ t

norm on any K ⋐ n × (, ∞) [Caffarelli-V

azquez 95]

▸ Hence, for a subsequence ε = ε j → +,

uε j → u locally uniformly in n × (, ∞)

▸ Any such u we call a limit solution of (P).

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Uniqueness of limit solutions

▸ Under certain geometric assumptions on u the limit solution is unique.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Uniqueness of limit solutions

▸ Under certain geometric assumptions on u the limit solution is unique.

Teorem ([ 02])

Let u be starshaped with respect to the origin in the sense u(λx) ≥ u(x) for any  < λ < , x ∈ n. Ten all limit solutions of (P) coincide with each other.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Uniqueness of limit solutions

▸ Under certain geometric assumptions on u the limit solution is unique.

Teorem ([ 02])

Let u be starshaped with respect to the origin in the sense u(λx) ≥ u(x) for any  < λ < , x ∈ n. Ten all limit solutions of (P) coincide with each other.

▸ Generally, however, there is no reason to expect uniqueness for general

initial data.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Uniqueness of limit solutions

▸ Under certain geometric assumptions on u the limit solution is unique.

Teorem ([ 02])

Let u be starshaped with respect to the origin in the sense u(λx) ≥ u(x) for any  < λ < , x ∈ n. Ten all limit solutions of (P) coincide with each other.

▸ Generally, however, there is no reason to expect uniqueness for general

initial data.

▸ [V

azquez 96] described the example where nonuniqueness of limit

solutions may occur.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example of Vazquez

▸ Let u be a radially symmetric classical

solution such that {u(⋅, t) > } = Br(t),  < t < Text,

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example of Vazquez

▸ Let u be a radially symmetric classical

solution such that {u(⋅, t) > } = Br(t),  < t < Text,

▸ Extiction time: u ≡  for t ≥ Text

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example of Vazquez

▸ Let u be a radially symmetric classical

solution such that {u(⋅, t) > } = Br(t),  < t < Text,

▸ Extiction time: u ≡  for t ≥ Text ▸ Maximal expansion:

r∗ = max

t∈[,Text] r(t)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example of Vazquez

▸ Let u be a radially symmetric classical

solution such that {u(⋅, t) > } = Br(t),  < t < Text,

▸ Extiction time: u ≡  for t ≥ Text ▸ Maximal expansion:

r∗ = max

t∈[,Text] r(t) ▸ Two hump initial data:

w(x) = u(x − r∗en) + u(x + r∗en)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Two hump initial data

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

▸ We start with a general theorem:

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

▸ We start with a general theorem:

Teorem ([-Yip 07])

For any u ∈ C(n) there exist minimal and maximal limit solutions u and u

• f (P) such that

u(x, t) ≤ u(x, t) ≤ u(x, t) for any limit solution of (P).

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

▸ We start with a general theorem:

Teorem ([-Yip 07])

For any u ∈ C(n) there exist minimal and maximal limit solutions u and u

• f (P) such that

u(x, t) ≤ u(x, t) ≤ u(x, t) for any limit solution of (P).

▸ Main difficulty: How to compare limits of uε over different sequences

ε j → +?

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

▸ We start with a general theorem:

Teorem ([-Yip 07])

For any u ∈ C(n) there exist minimal and maximal limit solutions u and u

• f (P) such that

u(x, t) ≤ u(x, t) ≤ u(x, t) for any limit solution of (P).

▸ Main difficulty: How to compare limits of uε over different sequences

ε j → +?

▸ Key facts:

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

▸ We start with a general theorem:

Teorem ([-Yip 07])

For any u ∈ C(n) there exist minimal and maximal limit solutions u and u

• f (P) such that

u(x, t) ≤ u(x, t) ≤ u(x, t) for any limit solution of (P).

▸ Main difficulty: How to compare limits of uε over different sequences

ε j → +?

▸ Key facts:

• 1. Every limit solution is a classical supersolution, i.e. ∣∇u∣ ≤  on ∂{u > }.
• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

▸ We start with a general theorem:

Teorem ([-Yip 07])

For any u ∈ C(n) there exist minimal and maximal limit solutions u and u

• f (P) such that

u(x, t) ≤ u(x, t) ≤ u(x, t) for any limit solution of (P).

▸ Main difficulty: How to compare limits of uε over different sequences

ε j → +?

▸ Key facts:

• 1. Every limit solution is a classical supersolution, i.e. ∣∇u∣ ≤  on ∂{u > }.
• 2. Every classical supersolution is a limit of supersolutions of (Pε) over any

sequence ε → +.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

Lemma (Strict comparison)

Let u, u be solutions of (P) with initial data u,, u,. Ten u, < u,

• n suppu,
• ⇒

u ≤ u in n × (, ∞)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

Lemma (Strict comparison)

Let u, u be solutions of (P) with initial data u,, u,. Ten u, < u,

• n suppu,
• ⇒

u ≤ u in n × (, ∞)

▸ We can now easily prove the existence of maximal/minimal solutions

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

Lemma (Strict comparison)

Let u, u be solutions of (P) with initial data u,, u,. Ten u, < u,

• n suppu,
• ⇒

u ≤ u in n × (, ∞)

▸ We can now easily prove the existence of maximal/minimal solutions ▸ Minimal solution: Let u j be a solution with initial data u, j such that

u, j ↗ u, suppu, j ⋐ {u > } Ten u j ↗ u

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Maximal and minimal limit solutions

Lemma (Strict comparison)

Let u, u be solutions of (P) with initial data u,, u,. Ten u, < u,

• n suppu,
• ⇒

u ≤ u in n × (, ∞)

▸ We can now easily prove the existence of maximal/minimal solutions ▸ Minimal solution: Let u j be a solution with initial data u, j such that

u, j ↗ u, suppu, j ⋐ {u > } Ten u j ↗ u

▸ Maximal solution: Let u j be a solution with initial data u, j such that

u, j ↘ u, suppu ⋐ {u, j > } Ten u j ↘ u

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example

Teorem ([-Yip 07])

Let w be the the two-hump initial data (described earlier) w(x) = u(x − r∗en) + u(x + r∗en). Ten w(x, t) = u(x − r∗en, t) + u(x + r∗en, t),

• n n × (, ∞)

w(x, t) > u(x − r∗en, t) + u(x + r∗en, t),

• n n × (t∗, t∗ + τ)

where t∗ is the time of maximal expansion and τ > 

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example

Teorem ([-Yip 07])

Let w be the the two-hump initial data (described earlier) w(x) = u(x − r∗en) + u(x + r∗en). Ten w(x, t) = u(x − r∗en, t) + u(x + r∗en, t),

• n n × (, ∞)

w(x, t) > u(x − r∗en, t) + u(x + r∗en, t),

• n n × (t∗, t∗ + τ)

where t∗ is the time of maximal expansion and τ > 

▸ Basically, one needs to construct a subsolution of (P) that “opens up” in a

small interval afer the two supports of u(⋅ ± r∗en, t) touch at t = t∗.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Nonuniqueness example

Teorem ([-Yip 07])

Let w be the the two-hump initial data (described earlier) w(x) = u(x − r∗en) + u(x + r∗en). Ten w(x, t) = u(x − r∗en, t) + u(x + r∗en, t),

• n n × (, ∞)

w(x, t) > u(x − r∗en, t) + u(x + r∗en, t),

• n n × (t∗, t∗ + τ)

where t∗ is the time of maximal expansion and τ > 

▸ Basically, one needs to construct a subsolution of (P) that “opens up” in a

small interval afer the two supports of u(⋅ ± r∗en, t) touch at t = t∗.

▸ We construct a subsolution in the form of the two circular solutions

connected with an “neck.”

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of subsolution

Idea:

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Alt-Caffarelli minimizers

▸ Minimize

J(v) ∶= ∫D (∣∇v∣ + χ{v>}) dx, v = v

• n ∂D.
• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Alt-Caffarelli minimizers

▸ Minimize

J(v) ∶= ∫D (∣∇v∣ + χ{v>}) dx, v = v

• n ∂D.

▸ Minimizers solve

∆v =  in {v > }, ∣∇v∣ = 

• n Γ ∶= ∂{v > } ∩ D,
• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Alt-Caffarelli minimizers

▸ Minimize

J(v) ∶= ∫D (∣∇v∣ + χ{v>}) dx, v = v

• n ∂D.

▸ Minimizers solve

∆v =  in {v > }, ∣∇v∣ = 

• n Γ ∶= ∂{v > } ∩ D,

v >  ∆v =  v ≡  ∣∇v∣ = 

• ✠
• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Catenoidal Alt-Caffarelli minimizers

Consider the minmizer v of J in B′

a × (−, )

with boundary values v = 

• n ∂B′

a × (−, )

v =  − κ

• n B′

a−η × {−} ∪ B′ a−η × {}

cylindrically symmetric in x′ variables v(x′, xn) = v(∣x′∣, xn).

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Catenoidal Alt-Caffarelli minimizers

Consider the minmizer v of J in B′

a × (−, )

with boundary values v = 

• n ∂B′

a × (−, )

v =  − κ

• n B′

a−η × {−} ∪ B′ a−η × {}

cylindrically symmetric in x′ variables v(x′, xn) = v(∣x′∣, xn).

Lemma

If κ < κ and a > a then

▸ ∂{v > } is smooth in B′ a × (−, ) ▸ ∂{v > } ⊂ (B′ a ∖ B′ ) × (−, )

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Catenoidal Alt-Caffarelli minimizers

▸ Sweep out by large spheres

Bλ(−̃ λen), Bλ(−̃ λen) ̃ λ = λ + √κ, λ = λ(a) by using strict comparison with explicit radually symmetric solutions

̃ λen −̃ λen λ ∣x′∣ = a ∣x′∣ = a x

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Catenoidal Alt-Caffarelli minimizers

▸ Sweep out by large spheres

Bλ(−̃ λen), Bλ(−̃ λen) ̃ λ = λ + √κ, λ = λ(a) by using strict comparison with explicit radually symmetric solutions

▸ Use density property:

 < c ≤ ∣{v = } ∩ Br(x)∣ ∣Br(x)∣ ≤  − c for any x ∈ ∂{v > }, Br(x) ⋐ B′

a × (−, )

̃ λen −̃ λen λ ∣x′∣ = a ∣x′∣ = a x

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of the “neck”

̂ vc = , ∣∇̂ vc∣ ≥  + κ/

• ✠

̂ vc ≤  − κ/

❆ ❆ ❑

B′

 × (− + η,  − η)

∆̂ vc = c S S S− S+ xn =  − η xn = −( − η)

▸ Take the catenoidal A-C

minimizer v.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of the “neck”

̂ vc = , ∣∇̂ vc∣ ≥  + κ/

• ✠

̂ vc ≤  − κ/

❆ ❆ ❑

B′

 × (− + η,  − η)

∆̂ vc = c S S S− S+ xn =  − η xn = −( − η)

▸ Take the catenoidal A-C

minimizer v.

▸ Smooth out corners of

V = {v > } outside ∣xn∣ ≤  − η.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of the “neck”

̂ vc = , ∣∇̂ vc∣ ≥  + κ/

• ✠

̂ vc ≤  − κ/

❆ ❆ ❑

B′

 × (− + η,  − η)

∆̂ vc = c S S S− S+ xn =  − η xn = −( − η)

▸ Take the catenoidal A-C

minimizer v.

▸ Smooth out corners of

V = {v > } outside ∣xn∣ ≤  − η.

▸ In the smoothened

domain ̂ V solve ∆̂ vc = c in ̂ V ̂ vc = ̂ v

• n ∂̂

V, where ̂ v = ( + κ/)v.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of the “neck”

̂ vc = , ∣∇̂ vc∣ ≥  + κ/

• ✠

̂ vc ≤  − κ/

❆ ❆ ❑

B′

 × (− + η,  − η)

∆̂ vc = c S S S− S+ xn =  − η xn = −( − η)

▸ Take the catenoidal A-C

minimizer v.

▸ Smooth out corners of

V = {v > } outside ∣xn∣ ≤  − η.

▸ In the smoothened

domain ̂ V solve ∆̂ vc = c in ̂ V ̂ vc = ̂ v

• n ∂̂

V, where ̂ v = ( + κ/)v.

▸ As c → + one has

̂ vc → ̂ v in C,α(̂ V)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of the “neck”

υ = , ∣∇υ∣ ≥  + κ/

• ✠

υ √αt ≤  − κ/

❆ ❆ ❑

∆υ − ∂tυ ≥  Σ(t) Σ(t) Σ−(t) Σ+(t)

xn √αt =  − η xn √αt = −( − η)

▸ Define

υ(x, t) ∶= √ αt ̂ vc ( x √αt ) in Υ ∶= {(x, t) ∶ x √αt ∈ ̂ V} .

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of the “neck”

υ = , ∣∇υ∣ ≥  + κ/

• ✠

υ √αt ≤  − κ/

❆ ❆ ❑

∆υ − ∂tυ ≥  Σ(t) Σ(t) Σ−(t) Σ+(t)

xn √αt =  − η xn √αt = −( − η)

▸ Define

υ(x, t) ∶= √ αt ̂ vc ( x √αt ) in Υ ∶= {(x, t) ∶ x √αt ∈ ̂ V} .

▸ Ten for α ≥ α(c), υ

satisfies ∆υ − ∂tυ ≥  in Υ

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of subsolution

u− Ω−(t) u+ Ω+(t) υ Υ(t) Σ(t) Σ+(t) Σ−(t)

ϕ(x, t) ∶= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ max{u−, υ, u+}, in Υ u−, in Ω− ∖ Υ u+, in Ω+ ∖ Υ , elsewhere

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of subsolution

▸ ϕ satisfies the free boundary condition ∣∇ϕ∣ ≥  on

∂{ϕ > } ∩ (n × (, τ)) in the sense liminf

x→x ϕ(x,t)>

ϕ(x, t) dist(x, ∂{ϕ(⋅, t) > }) ≥  for any x ∈ ∂{ϕ(⋅, t > },  < t < τ.

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of subsolution

▸ ϕ satisfies the free boundary condition ∣∇ϕ∣ ≥  on

∂{ϕ > } ∩ (n × (, τ)) in the sense liminf

x→x ϕ(x,t)>

ϕ(x, t) dist(x, ∂{ϕ(⋅, t) > }) ≥  for any x ∈ ∂{ϕ(⋅, t > },  < t < τ.

▸ Strict comparison works for ϕ against any limit solution of (P)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Construction of subsolution

▸ ϕ satisfies the free boundary condition ∣∇ϕ∣ ≥  on

∂{ϕ > } ∩ (n × (, τ)) in the sense liminf

x→x ϕ(x,t)>

ϕ(x, t) dist(x, ∂{ϕ(⋅, t) > }) ≥  for any x ∈ ∂{ϕ(⋅, t > },  < t < τ.

▸ Strict comparison works for ϕ against any limit solution of (P) ▸ Tis implies that the maximal limit solution w of (P) satisfies

w(x, t∗ + τ) ≥ ϕ(x, τ)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Proof of Nonuniqueness Teorem

▸ If wε is a solution of (Pε) such that

wε(⋅, t∗) > u−(⋅, t∗) + u+(⋅, t∗)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Proof of Nonuniqueness Teorem

▸ If wε is a solution of (Pε) such that

wε(⋅, t∗) > u−(⋅, t∗) + u+(⋅, t∗)

▸ Ten

wε(⋅, t∗) > ϕ(⋅, δε)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion

Proof of Nonuniqueness Teorem

▸ If wε is a solution of (Pε) such that

wε(⋅, t∗) > u−(⋅, t∗) + u+(⋅, t∗)

▸ Ten

wε(⋅, t∗) > ϕ(⋅, δε) wε(⋅, t∗ + τ) ≥ ϕ(⋅, τ + δε) ⇒ w(⋅, t∗ + τ) ≥ ϕ(⋅, τ)

• A. Petrosyan

Nonuniqeness in a free boundary problem from combustion