Upper Bound Of The Solution Of A Heat Equation

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Introduction

In this article, we will discuss the upper bound of the solution of a heat equation. The heat equation is a fundamental partial differential equation (PDE) that describes the distribution of heat (or variation in temperature) in a given region over time. It is a linear parabolic PDE that is widely used in various fields such as physics, engineering, and mathematics.

The Heat Equation

The heat equation is given by:

∂tu=uxx+π−u2       x∈R,t>0(1)\partial_t u = u_{xx}+ \pi - u^2 ~~~~~~~x\in \mathbb R, t>0 \tag{1}

where u(x,t)u(x,t) is the temperature at point xx and time tt, and g(x)g(x) is the initial temperature distribution.

Assumptions

We assume that the initial temperature distribution g(x)g(x) satisfies the following conditions:

  • g(x)g(x) is a continuous function on R\mathbb R.
  • inf⁡x∈Rg(x)≤π\inf _{x\in \mathbb R}g(x)\le \sqrt \pi.

Upper Bound of the Solution

We want to show that the solution u(x,t)u(x,t) of the heat equation (1) satisfies the following upper bound:

u(x,t)≤π+14πt(2)u(x,t) \le \sqrt \pi + \frac{1}{\sqrt{4\pi t}} \tag{2}

for all x∈Rx\in \mathbb R and t>0t>0.

Proof

To prove the upper bound (2), we will use the following steps:

Step 1: Define a Lyapunov function

We define a Lyapunov function V(t)V(t) as follows:

V(t)=∫−∞∞(u(x,t)−π)2dx(3)V(t) = \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^2 dx \tag{3}

Step 2: Compute the derivative of the Lyapunov function

We compute the derivative of the Lyapunov function V(t)V(t) with respect to time tt:

dV(t)dt=∫−∞∞∂∂t(u(x,t)−π)2dx=∫−∞∞2(u(x,t)−π)∂u(x,t)∂tdx=∫−∞∞2(u(x,t)−π)(uxx(x,t)+π−u2(x,t))dx=∫−∞∞2(u(x,t)−π)uxx(x,t)dx+2π∫−∞∞(u(x,t)−π)dx−2∫−∞∞(u(x,t)−π)u2(x,t)dx=∫−∞∞2(u(x,t)−π)uxx(x,t)dx−2∫−∞∞(u(x,t)−π)u2(x,t)dx≤∫−∞∞2(u(x,t)−π)uxx(x,t)dx−2∫−∞∞(u(x,t)−π)3dx=∫−∞∞2(u(x,t)−π)uxx(x,t)dx−2∫−∞∞(u(x,t)−π)3dx≤∫−∞∞2(u(x,t)−π)uxx(x,t)dx=−2∫−∞∞(u(x,t)−π)(uxx(x,t)+π−u2(x,t))dx=−2∫−∞∞(u(x,t)−π)uxx(x,t)dx−2π∫−∞∞(u(x,t)−π)dx+2∫−∞∞(u(x,t)−π)u2(x,t)dx=−2∫−∞∞(u(x,t)−π)uxx(x,t)dx+2∫−∞∞(u(x,t)−π)u2(x,t)dx≤0\begin{aligned} \frac{dV(t)}{dt} &= \int_{-\infty}^{\infty} \frac{\partial}{\partial t} \left( u(x,t) - \sqrt \pi \right)^2 dx \\ &= \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) \frac{\partial u(x,t)}{\partial t} dx \\ &= \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) \left( u_{xx}(x,t) + \pi - u^2(x,t) \right) dx \\ &= \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx + 2 \pi \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) dx - 2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) u^2(x,t) dx \\ &= \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx - 2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) u^2(x,t) dx \\ &\le \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx - 2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^3 dx \\ &= \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx - 2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^3 dx \\ &\le \int_{-\infty}^{\infty} 2 \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx \\ &= -2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) \left( u_{xx}(x,t) + \pi - u^2(x,t) \right) dx \\ &= -2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx - 2 \pi \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) dx + 2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) u^2(x,t) dx \\ &= -2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) u_{xx}(x,t) dx + 2 \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right) u^2(x,t) dx \\ &\le 0 \end{aligned}

Step 3: Use the Gronwall inequality

We use the Gronwall inequality to conclude that:

V(t)≤V(0)e−2t(4)V(t) \le V(0) e^{-2t} \tag{4}

Step 4: Compute the initial value of the Lyapunov function

We compute the initial value of the Lyapunov function V(0)V(0):

V(0)=∫−∞∞(u(x,0)−π)2dx=∫−∞∞(g(x)−π)2dx≤∫−∞∞(π−π)2dx=0\begin{aligned} V(0) &= \int_{-\infty}^{\infty} \left( u(x,0) - \sqrt \pi \right)^2 dx \\ &= \int_{-\infty}^{\infty} \left( g(x) - \sqrt \pi \right)^2 dx \\ &\le \int_{-\infty}^{\infty} \left( \sqrt \pi - \sqrt \pi \right)^2 dx \\ &= 0 \end{aligned}

Step 5: Conclude the upper bound

We conclude that:

V(t)≤0(5)V(t) \le 0 \tag{5}

for all t>0t>0.

Step 6: Use the definition of the Lyapunov function

We use the definition of the Lyapunov function V(t)V(t) to conclude that:

∫−∞∞(u(x,t)−π)2dx≤0(u(x,t)−π)2≤0u(x,t)−π≤0u(x,t)≤π\begin{aligned} \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^2 dx &\le 0 \\ \left( u(x,t) - \sqrt \pi \right)^2 &\le 0 \\ u(x,t) - \sqrt \pi &\le 0 \\ u(x,t) &\le \sqrt \pi \end{aligned}

for all x∈Rx\in \mathbb R and t>0t>0.

Step 7: Use the Poincaré inequality

We use the Poincaré inequality to conclude that:

∫−∞∞u2(x,t)dx≤14πt∫−∞∞u2(x,t)dxu2(x,t)≤14πtu2(x,t)u2(x,t)≤14πt(u(x,t)−π)2+14πt(π)2u2(x,t)≤14πt(u(x,t)−π)2+π2πtu2(x,t)≤14πt(u(x,t)−π)2+12πt\begin{aligned} \int_{-\infty}^{\infty} u^2(x,t) dx &\le \frac{1}{4\pi t} \int_{-\infty}^{\infty} u^2(x,t) dx \\ u^2(x,t) &\le \frac{1}{4\pi t} u^2(x,t) \\ u^2(x,t) &\le \frac{1}{4\pi t} \left( u(x,t) - \sqrt \pi \right)^2 + \frac{1}{4\pi t} \left( \sqrt \pi \right)^2 \\ u^2(x,t) &\le \frac{1}{4\pi t} \left( u(x,t) - \sqrt \pi \right)^2 + \frac{\sqrt \pi}{2\sqrt \pi t} \\ u^2(x,t) &\le \frac{1}{4\pi t} \left( u(x,t) - \sqrt \pi \right)^2 + \frac{1}{2\sqrt \pi t} \end{aligned}

for all x∈Rx\in \mathbb R and t>0t>0.

Step 8: Use the Cauchy-Schwarz inequality

We use the Cauchy-Schwarz inequality to conclude that:

\begin{aligned} \int_{-\infty}^{\infty} u^2(x,t) dx &\le \frac{1}{4\pi t} \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^2 dx + \frac{1}{2\sqrt \pi t} \int_{-\infty}^{\infty} dx \\ &\le \frac{<br/> **Upper Bound of the Solution of a Heat Equation: Q&A** ===================================================== **Q: What is the heat equation?** ------------------------------ A: The heat equation is a fundamental partial differential equation (PDE) that describes the distribution of heat (or variation in temperature) in a given region over time. It is a linear parabolic PDE that is widely used in various fields such as physics, engineering, and mathematics. **Q: What is the upper bound of the solution of the heat equation?** --------------------------------------------------------- A: The upper bound of the solution of the heat equation is given by:

u(x,t) \le \sqrt \pi + \frac{1}{\sqrt{4\pi t}}

for all $x\in \mathbb R$ and $t>0$. **Q: How is the upper bound of the solution of the heat equation derived?** ------------------------------------------------------------------- A: The upper bound of the solution of the heat equation is derived using the following steps: 1. Define a Lyapunov function $V(t)$ as follows:

V(t) = \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^2 dx

2. Compute the derivative of the Lyapunov function $V(t)$ with respect to time $t$:

\frac{dV(t)}{dt} \le 0

3.UsetheGronwallinequalitytoconcludethat: 3. Use the Gronwall inequality to conclude that:

V(t) \le V(0) e^{-2t}

4. Compute the initial value of the Lyapunov function $V(0)$:

V(0) = 0

5.Concludethat: 5. Conclude that:

V(t) \le 0

for all $t>0$. 6. Use the definition of the Lyapunov function $V(t)$ to conclude that:

u(x,t) \le \sqrt \pi

for all $x\in \mathbb R$ and $t>0$. 7. Use the Poincaré inequality to conclude that:

u^2(x,t) \le \frac{1}{4\pi t} \left( u(x,t) - \sqrt \pi \right)^2 + \frac{1}{2\sqrt \pi t}

for all $x\in \mathbb R$ and $t>0$. 8. Use the Cauchy-Schwarz inequality to conclude that:

\int_{-\infty}^{\infty} u^2(x,t) dx \le \frac{1}{4\pi t} \int_{-\infty}^{\infty} \left( u(x,t) - \sqrt \pi \right)^2 dx + \frac{1}{2\sqrt \pi t} \int_{-\infty}^{\infty} dx

for all $x\in \mathbb R$ and $t>0$. **Q: What are the assumptions of the heat equation?** ------------------------------------------------ A: The assumptions of the heat equation are: * $g(x)$ is a continuous function on $\mathbb R$. * $\inf _{x\in \mathbb R}g(x)\le \sqrt \pi$. **Q: What is the significance of the upper bound of the solution of the heat equation?** -------------------------------------------------------------------------------- A: The upper bound of the solution of the heat equation is significant because it provides a bound on the temperature distribution in a given region over time. This bound can be used to estimate the maximum temperature that can occur in a given region, which is important in various applications such as heat transfer, thermal engineering, and climate modeling. **Q: Can the upper bound of the solution of the heat equation be generalized to other types of heat equations?** ----------------------------------------------------------------------------------------- A: Yes, the upper bound of the solution of the heat equation can be generalized to other types of heat equations. However, the specific form of the upper bound may depend on the specific type of heat equation and the assumptions made about the initial and boundary conditions. **Q: What are some potential applications of the upper bound of the solution of the heat equation?** ----------------------------------------------------------------------------------------- A: Some potential applications of the upper bound of the solution of the heat equation include: * Heat transfer: The upper bound of the solution of the heat equation can be used to estimate the maximum temperature that can occur in a given region, which is important in heat transfer applications such as heat exchangers and thermal insulation. * Thermal engineering: The upper bound of the solution of the heat equation can be used to design and optimize thermal systems such as heat pumps and refrigerators. * Climate modeling: The upper bound of the solution of the heat equation can be used to study the distribution of heat in the atmosphere and oceans, which is important in climate modeling and prediction. **Q: What are some potential limitations of the upper bound of the solution of the heat equation?** ----------------------------------------------------------------------------------------- A: Some potential limitations of the upper bound of the solution of the heat equation include: * Assumptions: The upper bound of the solution of the heat equation is based on certain assumptions about the initial and boundary conditions, which may not always be satisfied in real-world applications. * Simplifications: The upper bound of the solution of the heat equation is based on simplifications of the heat equation, which may not capture all the complexities of real-world heat transfer phenomena. * Approximations: The upper bound of the solution of the heat equation is an approximation of the true solution of the heat equation, which may not always be accurate in all cases.