Proving Exponential Decay of Solution u_t w/o Poincaré Inequality

[onie mti]

I tried solving this problem, i uploaded how far I went. please correct if I am wrong View attachment 2179View attachment 2180

Let u be the solution of the equation
u_t (x,t) - Kuxx (x,t) + ku(x,t) = 0; 0 < x < ℓ; t > 0;
under zero-flux boundary conditions and the initial condition u(x,0) = a(x).
Prove that ∥u(
,t)∥ decays exponentially as t tends to infinity even if A ̸= 0. The Poincare
inequality is not required for the proof.

 

[gtoguy97]

Proof:Let us assume that u(x,t) is the solution of the equationu_t (x,t) - Kuxx (x,t) + ku(x,t) = 0; 0 < x < ℓ; t > 0;under zero-flux boundary conditions and the initial condition u(x,0) = a(x).We will prove that ∥u(
,t)∥ decays exponentially as t tends to infinity even if A ̸= 0.First, we note that the equation can be written asu_t (x,t) = Kuxx (x,t) - ku(x,t).Now, we take the norm of both sides of the equation:∥u_t (x,t)∥ = ∥Kuxx (x,t) - ku(x,t)∥Using the triangle inequality, we get:∥u_t (x,t)∥ ≤ ∥Kuxx (x,t)∥ + ∥ku(x,t)∥.Next, we use the fact that ∥u(x,t)∥≤∥u(x,0)∥ to get:∥u_t (x,t)∥ ≤ ∥Kuxx (x,t)∥ + k∥u(x,0)∥.Now, we take the integral of both sides of the equation over the interval [0,ℓ]:∫_0^ℓ ∥u_t (x,t)∥dx ≤ ∫_0^ℓ ∥Kuxx (x,t)∥dx + k∫_0^ℓ∥u(x,0)∥dx.Since we are considering a zero-flux boundary condition, we have∫_0^ℓ ∥u_t (x,t)∥dx = 0.Therefore, we get0 ≤ ∫_0^ℓ ∥Kuxx (x,t)∥dx + k∫_0^ℓ