A Criterion for Blow Up in Finite Time of a System of 1-Dimensional Reaction-Diffusion Equations

Abstract

We give a criterion for blow up in finite time of the system of semilinear partial differential equations $\frac {\partial u_{i}(t,x)}{\partial t}=\frac {1}{2}\frac {\partial ^{2}u_{i}\left (t,x\right )}{\partial x^{2}}+\frac {\varphi ^{\prime }_{i}\left (x\right )} {\varphi _{i}\left (x\right )}\frac {\partial u_{i}\left (t,x\right )}{\partial x}+u_{j}^{1+\beta _{i}}\left (t,x\right )$, t > 0, $x\in \mathbb {R}$, with initial values of the form $u_{i}\left (0,x\right )={h_{i}\left (x\right )}/{\varphi _{i}\left (x\right )}$, where $0<\varphi _{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\cap C^{2}\left (\mathbb {R}\right )$, $0\leq h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )$, β _i  > 0 and i = 1, 2, j = 3 − i. Moreover, we find an upper bound T ^∗ for the blowup time of such system which depends both on the initial values f ₁, f ₂, and the measures $\mu _i(\mathrm {d} x)=\varphi _i^2(x)\,\mathrm {d} x$, i = 1, 2.

1 Introduction

Consider the semilinear partial differential equation

$$\displaystyle \begin{aligned} \frac{\partial u(t,x)}{\partial t} =\frac{1}{2}\frac{\partial^{2} u(t,x)}{\partial x^{2}} +\frac{\varphi'\left(x\right)}{\varphi\left(x\right)}\frac{\partial u(t,x)}{\partial x} +u^{1+\beta}\left(t,x\right), \quad t>0, \quad x\in\mathbb{R}, \end{aligned} $$

(1)

where β > 0, $\varphi \in C^2(\mathbb {R})$ is a square-integrable, strictly positive function, and the initial value is of the form u(0, x) = h(x)∕φ(x) with $h\in L^2(\mathbb {R},\mathrm {d} x)$ and φ(x) = dφ(x)∕dx. Setting $\varphi (x)=e^{-x^2/2}$ in (1) it becomes

$$\displaystyle \begin{aligned} \frac{\partial u(t,x)}{\partial t} =L^{\varphi}u(t,x) +u^{1+\beta}\left(t,x\right), \quad t>0, \quad x\in\mathbb{R}, \end{aligned}$$

where $L^{\varphi }:= \frac {1}{2}\frac {\partial ^{2} }{\partial x^{2}}-x\frac {\partial }{\partial x}$ is the infinitesimal generator of the Ornstein-Uhlenbeck semigroup {T _t , t ≥ 0}. Using essentially Jensen’s inequality and the fact that the measure μ(dx) = φ ²(x) dx is invariant for {T _t , t ≥ 0}, in [8] we were able to prove that Eq. (1) exhibits blow up in finite time for any nontrivial initial value of the form u(0, x) = h(x)∕φ(x), $x\in \mathbb {R}$.

Motivated by this example, in this note we provide a criterion for explosion in finite time of positive mild solutions of the 1-dimensional semilinear system

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\partial u_{1}(t,x)}{\partial t} &\displaystyle =&\displaystyle \frac{1}{2}\frac{\partial^{2} u_{1}(t,x)}{\partial x^{2}} +\frac{\varphi^{\prime}_{1}\left(x\right)}{\varphi_{1}\left(x\right)}\frac{\partial u_{1}(t,x)}{\partial x} +u_{2}^{1+\beta_{1}}\left(t,x\right), \quad t>0, \quad x\in\mathbb{R},\\ {} \frac{\partial u_{2}(t,x)}{\partial t} &\displaystyle =&\displaystyle \frac{1}{2}\frac{\partial^{2} u_{2}(t,x)}{\partial x^{2}} +\frac{\varphi^{\prime}_{2}\left(x\right)}{\varphi_{2}\left(x\right)}\frac{\partial u_{2}(t,x)}{\partial x} +u_{1}^{1+\beta_{2}}\left(t,x\right), \quad t>0, \quad x\in\mathbb{R},\\ u_{i}(0,x) &\displaystyle = &\displaystyle f_{i}(x),\quad x\in\mathbb{R},\quad i=1,2, \end{array} \end{aligned} $$

(2)

where β ₁, β ₂ > 0 are constants, f ₁, f ₂ are nonnegative functions and $\varphi _1,\varphi _2\in C^2(\mathbb {R})\cap L^2(\mathbb {R},\mathrm {d} x)$ are strictly positive. Semilinear systems of this type have been investigated intensively in last years, starting with the pioneering work of Galaktionov et al. [4] (see also [2, 3, 5, 7, 9] and the review papers [1, 6]). This kind of systems arise as simplified models of the process of diffusion of heat and burning in a two-component continuous media, where u ₁ and u ₂ represent the temperatures of the two reactant components.

Recall that a pair (u ₁, u ₂) of measurable functions is termed mild solution of system (2) if it solves the system of integral equations

$$\displaystyle \begin{aligned} u_i(t,x)=T^{i}_t\left(f_i(x)\right)+\int_0^t T^{i}_{t-s}\left(u_j^{1+\beta_i}(s,x)\right)\,\mathrm{d} s,\quad t\ge0,\quad x\in\mathbb{R}, \end{aligned} $$

(3)

where i = 1, 2, j = 3 − i and $\{T^i_t$, t ≥ 0} is the semigroup of continuous linear operators on $L^{\infty }(\mathbb {R},\mathrm {d} x)$ having infinitesimal generator

$$\displaystyle \begin{aligned} L^{\varphi_i}=\frac{1}{2}\frac{\partial^2}{\partial x^2} + \frac{\varphi^{\prime}_i}{\varphi_i}\frac{\partial}{\partial x};\quad i=1,2. \end{aligned}$$

If there exists $T\,{\in }\,\left (0,\infty \right )$ such that $\left \Vert u_{1}\left (t,\cdot \right )\right \Vert _{L^{\infty }\left (\mathbb {R},\mathrm {d} x\right )}\,{=}\,\infty$ or $\left \Vert u_{2}\left (t,\cdot \right )\right \Vert _{L^{\infty }\left (\mathbb {R},\mathrm {d} x\right )}=\infty$ for all t ≥ T, then it is said that $\left (u_{1},u_{2}\right )$ blows up (or explodes) in finite time, and in this case the infimum of such T’s is called the blow up time (or the explosion time) of $\left (u_{1},u_{2}\right )$.

Notice that for any $g\in L^{\infty }(\mathbb {R},\mathrm {d} x)$ and i = 1, 2,

$$\displaystyle \begin{aligned} T^i_t(g(x))=\mathbb{E}\left[g\left(X_t^{x,i}\right)\right] ,\quad t\ge0,\quad x\in\mathbb{R}, \end{aligned}$$

where $\{X_t^{x,i}$, t ≥ 0} is the unique strong solution of the stochastic differential equation

$$\displaystyle \begin{aligned} Y_t=x+B_t+\int_0^t\frac{\varphi^{\prime}_i}{\varphi_i}\left(Y_s\right)\,\mathrm{d} s,\quad t\ge0,\quad x\in\mathbb{R}; \end{aligned}$$

here {B _t , t ≥ 0} is a standard 1-dimensional Brownian motion. It turns out that under our assumptions both processes $\{X_t^{x,i}$, t ≥ 0}, i = 1, 2, are recurrent and, moreover, possess corresponding invariant measures

$$\displaystyle \begin{aligned} \mu_i(\mathrm{d} x)=\varphi_i^2(x)\,\mathrm{d} x,\quad i=1,2. \end{aligned} $$

(4)

The intuitive explanation of the blow up phenomenon in non-linear heat equations of the archetype

$$\displaystyle \begin{aligned} \frac{\partial u}{\partial t} = {\mathcal{A}}u + u^{1+\beta};\quad u(0)=f\ge0, \end{aligned}$$

where β > 0 and ${\mathcal {A}}$ is the generator of a strong Markov process on a locally compact space, is that if the initial value f is “small” then the tendency of the solution to blow up (which it would do if u ^1+β were the only term in the left-hand side of the equation) can be inhibited by the dissipative effect of the migration with generator ${\mathcal {A}}$; see e.g. [6, 9] or [10]. In view of the ergodicity of the processes $\{X_t^{x,i}$, t ≥ 0}, i = 1, 2, the mild solution of (2) should therefore blow up in finite time, at least for certain non-trivial positive initial values f _i , i = 1, 2.

In this work we give conditions which imply blow up in finite time of system (2) under the assumption that φ ₁∕φ ₂ is a strictly positive bounded function such that $\displaystyle \inf _{x\in \mathbb {R}}\{\varphi _{1}\left (x\right )/\varphi _{2}\left (x\right )\}>0$, and the initial values are of the form f _i  = h _i ∕φ _i , where $h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )$, i = 1, 2. We distinguish two cases: if β ₁ = β ₂ we show that any non-trivial positive mild solution of (2) blows up in finite time. If β ₁ ≠ β ₂ we prove that a condition on the “sizes” of f ₁ and f ₂ and on the measures μ ₁, μ ₂ of the form

$$\displaystyle \begin{aligned} \int f_1\,\mathrm{d}\mu_1+\int f_2\,\mathrm{d}\mu_2 > c_0, \end{aligned}$$

(where the constant c ₀ > 0 is determined by the system parameters) already implies finite time explosion of (2); see Theorem 2 below. Moreover, we find an upper bound T ^∗ for the blowup time of system (2) which depends both on the initial values f ₁, f ₂, and the invariant measures (4). Our setting allows us to consider a wide range of choices for φ ₁ and φ ₂, for instance

$$\displaystyle \begin{aligned}\varphi_{1}\left(x\right)=\left(\sin\left(x\right)+2\right)\varphi_{2}\left(x\right) \mbox{ with }\varphi_{2}\left(x\right)=e^{-x^{2}/2}, \end{aligned}$$

or else

$$\displaystyle \begin{aligned}\varphi_{1}\left(x\right)=\left(e^{-x^{2}/2}+1\right)\varphi_{2}\left(x\right) \mbox{ with } \varphi_{2}\left(x\right)={1}/{(1+x^{2})}. \end{aligned}$$

In these two cases the functions h _i , i = 1, 2, can be chosen of the form $h_{i}\left (x\right )={P_{i}\left (\left |x\right |\right )}/{Q_{i}\left (\left |x\right |\right )}$, where P _i , Q _i are polynomial functions with non-negative coefficients such that their degrees satisfy $2\leq \mathrm {deg}\left (Q_{i}\right )-\mathrm {deg}\left (P_{i}\right )$, and $Q_{i}\left (0\right )>0$.

In the next section we prove existence and uniqueness of local mild solutions of (2) using the classical fixed-point argument, adapted to our context. Our main result, Theorem 2, is stated and proved in Sect. 3.

2 Local Existence and Uniqueness of Mild Solutions

Our proof of existence, uniqueness and positiveness of mild solutions of system (2) is based on [14, Theorem 2.1], (see also [12, Theorem 2.1], [15, Theorem 3], [7, Theorem 2] or [11, Theorem 1]).

For each $\tau \in \left (0,\infty \right )$ we define the set

$$\displaystyle \begin{aligned} E_{\tau}:=\left\{\left(u_{1},u_{2}\right)|u_{1},u_{2}: \left[0,\tau\right]\to L^{\infty}\left(\mathbb{R},\mathrm{d} x\right), \left|\left|\left| \left(u_{1},u_{2}\right)\right| \right|\right|<\infty\right\}, \end{aligned}$$

where

$$\displaystyle \begin{aligned} \left|\left|\left| \left(u_{1},u_{2}\right)\right| \right|\right|:=\sup_{t\in\left[0,\tau\right]}\left\{ \left\Vert u_{1}\left(t,\cdot\right)\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}+ \left\Vert u_{2}\left(t,\cdot\right)\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right\}. \end{aligned}$$

Then $\left (E_{\tau },\left |\left |\left | \cdot \right | \right |\right |\right )$ is a Banach space and the sets

$$\displaystyle \begin{aligned} &P_{\tau}:=\left\{\left(u_{1},u_{2}\right)\in E_{\tau}:u_{1}\geq0,u_{2}\geq 0\right\} \quad \mbox{and}\\ & B_{R}:=\left\{\left(u_{1},u_{2}\right)\in E_{\tau}:\left|\left|\left|\left(u_{1},u_{2}\right)\right|\right|\right|\leq R\right\} \end{aligned} $$

are closed subsets of E _τ for any $R\in \left (0,\infty \right )$. Therefore $\left (P_{\tau }\cap B_{R},\left |\left |\left | \cdot \right | \right |\right |\right )$ is a Banach space for all $\tau , R\in \left (0,\infty \right )$.

Theorem 1

There exist $\tau ,R\in \left (0,\infty \right )$ such that system (2) has a unique positive mild solution in P _τ  ∩ B _R .

Proof

We will prove that the operator Ψ : P _τ  ∩ B _R  → P _τ  ∩ B _R defined by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \Psi\left(\left(u_{1}\left(t,x\right),u_{2}\left(t,x\right)\right)\right) &\displaystyle =&\displaystyle \left(T_{t}^{1}\left(f_{1}\left(x\right)\right)+ \int_{0}^{t}T_{t-s}^{1}\left(u_{2}^{1+\beta_{1}}\left(s,x\right)\right)\mathrm{d} s,\right.\\ &\displaystyle &\displaystyle \left.T_{t}^{2}\left(f_{2}\left(x\right)\right) + \int_{0}^{t}T_{t-s}^{2}\left(u_{1}^{1+\beta_{2}}\left(s,x\right)\right) \mathrm{d} s\right), \end{array} \end{aligned} $$

is a contraction for certain $\tau ,R\in \left (0,\infty \right )$. We start by verifying that Ψ is in fact an operator from P _τ  ∩ B _R onto P _τ  ∩ B _R for suitably chosen $\tau ,R\in \left (0,\infty \right )$. Let $\tau _{0},R_{0}\in \left (0,\infty \right )$ be such that

$$\displaystyle \begin{aligned} &R_{0}>\left(\left\Vert f_{1}\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}+ \left\Vert f_{2}\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right)\;\mbox{and}\\ &\tau_{0}\leq\frac{R_{0}-\left(\left\Vert f_{1}\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}+ \left\Vert f_{2}\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right)}{R_{0}^{1+\beta_{1}}+R_{0}^{1+\beta_{2}}}. \end{aligned} $$

If $\left (u_{1},u_{2}\right )\in P_{\tau _{0}}\cap B_{R_{0}}$ then $\Psi \left (\left (u_{1},u_{2}\right )\right )$ has positive components due to the definition of Ψ and the fact that u ₁, u ₂ ≥ 0. Hence

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left|\left|\left| \Psi\left(\left(u_{1},u_{2}\right)\right)\right| \right|\right| &\displaystyle =&\displaystyle \sup_{t\in\left[0,\tau_{0}\right]}\left\{\left\Vert T_{t}^{1}\left(f_{1} \left(\cdot\right)\right) + \int_{0}^{t}T_{t-s}^{1}\left(u_{2}^{1+\beta_{1}}\left(s,\cdot\right)\right) \mathrm{d} s\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right.\\ &\displaystyle &\displaystyle +\left.\left\Vert T_{t}^{2}\left(f_{2}\left(\cdot\right)\right) + \int_{0}^{t}T_{t-s}^{2}\left(u_{1}^{1+\beta_{2}}\left(s,\cdot\right)\right) \mathrm{d} s\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right\}\\ &\displaystyle \leq&\displaystyle \left\Vert f_{1}\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}+\left\Vert f_{2}\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}+ \tau_{0}\left(R_{0}^{1+\beta_{1}}+R_{0}^{1+\beta_{2}}\right), \end{array} \end{aligned} $$

where we have used the contraction property of the operators $T^i_t$, i = 1, 2, to obtain the last inequality. It follows that $\left |\left |\left | \Psi \left (\left (u_{1},u_{2}\right )\right )\right | \right |\right |\leq R_{0}$, i.e., Ψ is an operator from $P_{\tau _{0}}\cap B_{R_{0}}$ onto itself.

In order to prove the contraction property of Ψ we choose τ ₀ as above in such a way that

$$\displaystyle \begin{aligned} \displaystyle\max_{i\in\left\{1,2\right\}}\left\{\left(1+\beta_{i}\right)R_{0}^{\beta_{i}}\right\}\tau_{0} \in\left(0,1\right). \end{aligned} $$

(5)

Let $\left (u_{1},u_{2}\right ),\left (\hat {u}_{1},\hat {u}_{2}\right )\in P_{\tau _{0}}\cap B_{R_{0}}$. Using again the contraction property of the operators $T^i_t$, i = 1, 2, and the well-known inequality $\left |a^{p}- b^{p}\right |\leq p\left (a\vee b\right )^{p-1}\left |a-b\right |$, which holds for all a, b > 0 and p ≥ 1, we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left|\left|\left| \Psi\left(\left(u_{1},u_{2}\right)\right)-\Psi\left(\left(\hat{u}_{1},\hat{u}_{2}\right)\right)\right| \right|\right|\\ &\displaystyle =&\displaystyle \sup_{t\in\left[0,\tau_{0}\right]}\left\{\left\Vert \int_{0}^{t}T^{1}_{t-s} \left(u_{2}^{1+\beta_{1}}\left(s,\cdot\right)- \hat{u}_{2}^{1+\beta_{1}}\left(s,\cdot\right)\right)\mathrm{d} s\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right.\\ &\displaystyle &\displaystyle +\left.\left\Vert \int_{0}^{t}T^{2}_{t-s}\left(u_{1}^{1+\beta_{2}}\left(s,\cdot\right)- \hat{u}_{1}^{1+\beta_{2}}\left(s,\cdot\right)\right)\mathrm{d} s\right\Vert _{L^{\infty}\left(\mathbb{R},\mathrm{d} x\right)}\right\}\\ &\displaystyle \leq&\displaystyle \sup_{t\in\left[0,\tau_{0}\right]} \int_{0}^{t}\left\Vert u_{2}^{1+\beta_{1}}\left(s,\cdot\right)-\hat{u}_{2}^{1+\beta_{1}}\left(s,\cdot\right)\right\Vert _{ L^\infty\left(\mathbb{R},\mathrm{d} x\right)}\mathrm{d} s\\ &\displaystyle &\displaystyle +\sup_{t\in\left[0,\tau_{0}\right]} \int_{0}^{t}\left\Vert u_{1}^{1+\beta_{2}}\left(s,\cdot\right)-\hat{u}_{1}^{1+\beta_{2}}\left(s,\cdot\right)\right\Vert _{ L^\infty\left(\mathbb{R},\mathrm{d} x\right)}\mathrm{d} s\\ &\displaystyle \leq&\displaystyle \left(1+\beta_{1}\right)R_{0}^{\beta_{1}} \int_{0}^{\tau_{0}}\left\Vert u_{2}\left(s,\cdot\right)-\hat{u}_{2}\left(s,\cdot\right)\right\Vert _{L^\infty\left(\mathbb{R},\mathrm{d} x\right)}\mathrm{d} s\\ &\displaystyle &\displaystyle +\left(1+\beta_{2}\right)R_{0}^{\beta_{2}} \int_{0}^{\tau_{0}}\left\Vert u_{1}\left(s,\cdot\right)- \hat{u}_{1}\left(s,\cdot\right)\right\Vert _{L^\infty\left(\mathbb{R},\mathrm{d} x\right)}\mathrm{d} s\\ &\displaystyle \leq&\displaystyle \max_{i\in\left\{1,2\right\}}\left\{\left(1+\beta_{i}\right)R_{0}^{\beta_{i}}\right\}\tau_{0} \left|\left|\left| \left(u_{1},u_{2}\right)-\left(\hat{u}_{1},\hat{u}_{2}\right)\right| \right|\right|. \end{array} \end{aligned} $$

From the last inequality we conclude, due to (5), that Ψ is a contraction in $P_{\tau _{0}}\cap B_{R_{0}}$. It follows from the Banach fixed-point theorem that Ψ has a unique fixed point in $P_{\tau _{0}}\cap B_{R_{0}}$, which is the unique mild solution of system (2). □

3 A Condition for Blowup in Finite Time

Our main result is the following

Theorem 2

Let $\varphi _{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\cap C^{2}\left (\mathbb {R}\right )$ be a strictly positive function and assume that the initial value f _i admits the representation

$$\displaystyle \begin{aligned} f_{i}\left(x\right):=\frac{h_{i}\left(x\right)}{\varphi_{i}\left(x\right)}\geq 0,\quad x\in\mathbb{R}, \end{aligned} $$

(6)

for some positive nontrivial $h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )$ , i = 1, 2. Suppose in addition that there exist strictly positive constants k ₁, k ₂ such that

$$\displaystyle \begin{aligned} k_{1}\leq\frac{\varphi_{1}\left(x\right)}{\varphi_{2}\left(x\right)}\leq k_{2},\quad x\in\mathbb{R}.\end{aligned} $$

(7)

• 1. Assume that β ₁ = β ₂. Then any non-trivial positive mild solution $\left (u_{1},u_{2}\right )$ of system (2) blows up in finite time.

• 2. Assume that β ₁ > β ₂. Let $A_{0}:=\left (\frac {1+\beta _{2}}{1+\beta _{1}}\right )^ {\frac {1+\beta _{2}}{\beta _{1}-\beta _{2}}}\frac {\beta _{1}-\beta _{2}}{1+\beta _{1}}$ and suppose that

  $$\displaystyle \begin{aligned} \int_{\mathbb{R}}f_{1}\left(x\right)\mu_{1}\left(\mathrm{d} x\right) +\int_{\mathbb{R}}f_{2}\left(x\right)\mu_{2}\left(\mathrm{d} x\right)> 2^{\frac{\beta_{2}}{1+\beta_{2}}}A^{\frac{1}{1+\beta_{2}}}_{0}. \end{aligned} $$

  (8)

  Then any mild solution $\left (u_{1},u_{2}\right )$ of system (2) blows up in finite time.

Proof

Let $\left (u_{1},u_{2}\right )$ be a mild solution of system (2). We denote

$$\displaystyle \begin{aligned} w_{i}\left(t,x\right):=\varphi_{i}\left(x\right)u_{i}\left(t,x\right), \quad t\geq0, \quad x\in\mathbb{R}. \end{aligned}$$

Multiplying both sides of (3) by φ _i yields

$$\displaystyle \begin{aligned} w_{i}\left(t,x\right)=\varphi_{i}\left(x\right)T_{t}^{i}\left(\frac{h_{i}}{\varphi_{i}}\left(x\right)\right)+ \int_{0}^{t}\varphi_{i}\left(x\right)T_{t-s}^{i}\left( w_{3-i}^{1+\beta_{i}}\left(s,x\right)\varphi_{3-i}^{-\left(1+\beta_{i}\right)}\left(x\right)\right)\mathrm{d} s. \end{aligned} $$

(9)

Since the function $g_{i}\left (x\right ):=\varphi _{i}^{2}\left (x\right )$ satisfies the differential equation

$$\displaystyle \begin{aligned} \frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}g_{i}\left(x\right)-\frac{\partial}{\partial x} \left(g_{i}\left(x\right)\frac{\varphi^{\prime}_{i}\left(x\right)}{\varphi_{i}\left(x\right)}\right)=0,\quad x\in\mathbb{R}, \end{aligned}$$

it follows that $\mu _{i}\left (\mathrm {d} x\right )= \varphi _{i}^{2}\left (x\right )\mathrm {d} x$ is invariant for the semigroup $\left \{T_{t}^{i},\ t\geq 0\right \}$. Let us write $\mathbb {E}^{i}\left [f\right ]:=\int _{\mathbb {R}}f\left (x\right )\varphi _{i}\left (x\right )\mathrm {d} x$. Due to (9) this implies that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbb{E}^{i}\left[w_{i}\left(t,\cdot\right)\right] &\displaystyle =&\displaystyle \mathbb{E}^{i}\left[h_{i}\left(\cdot\right)\right]+ \int_{0}^{t}\mathbb{E}^{i}\left[w_{3-i}^{1+\beta_{i}}\left(s,\cdot\right)\varphi_{i}\left(\cdot\right) \varphi_{3-i}^{-\left(1+\beta_{i}\right)}\left(\cdot\right)\right]\mathrm{d} s.{} \end{array} \end{aligned} $$

(10)

Define $\displaystyle a:=\min \left \{k_{1}^{2},\frac {1}{k_{2}^{2}}\right \}$. From assumption (7) we get $\frac {\displaystyle \varphi _{i}^{2}\left (x\right )}{\displaystyle \varphi _{3-i}^{2}\left (x\right )}\geq a$ for all $x\in \mathbb {R}$ and i = 1, 2. Therefore

$$\displaystyle \begin{aligned} & \mathbb{E}^{i}\left[w_{3-i}^{1+\beta_{i}}\left(s,\cdot\right)\varphi_{i}\left(\cdot\right) \varphi_{3-i}^{-\left(1+\beta_{i}\right)}\left(\cdot\right)\right]\\ &\quad =\int_{\mathbb{R}}\left(\frac{w_{3-i}\left(s,x\right)}{\varphi_{3-i}\left(x\right)}\right)^{1+\beta_{i}} \varphi_{i}^{2}\left(x\right)\mathrm{d} x\\ &\quad \geq a\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{2} \int_{\mathbb{R}}\left(\frac{w_{3-i}\left(s,x\right)}{\varphi_{3-i}\left(x\right)}\right)^{1+\beta_{i}} \frac{\varphi_{3-i}^{2}\left(x\right)}{\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{2}}\mathrm{d} x \\\noalign{} &\quad \geq a\frac{\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{2}} {\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{2+2\beta_{i}}} \left(\int_{\mathbb{R}}\frac{w_{3-i}\left(s,x\right)}{\varphi_{3-i}\left(x\right)} \varphi_{3-i}^{2}\left(x\right)\mathrm{d} x\right) ^{1+\beta_{i}}\\ {} &\quad = a\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{-2\beta_{i}} \left(\mathbb{E}^{3-i}\left[w_{3-i}\left(s,\cdot\right)\right]\right)^{1+\beta_{i}}, \end{aligned} $$

(11)

where we have used Jensen’s inequality to obtain the last inequality. Plugging (11) into (10) renders

$$\displaystyle \begin{aligned} \mathbb{E}^{i}\left[w_{i}\left(t,\cdot\right)\right]\geq\mathbb{E}^{i}\left[h_{i}\left(\cdot\right)\right]+ a\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{-2\beta_{i}}\int_{0}^{t} \left(\mathbb{E}^{3-i}\left[w_{3-i}\left(s,\cdot\right)\right]\right)^{1+\beta_{i}}\mathrm{d} s. \end{aligned} $$

(12)

Let $y_{i}\left (t\right )$ be the solution of the system

$$\displaystyle \begin{aligned} \begin{array}{rcl} y^{\prime}_{i}\left(t\right) &\displaystyle =&\displaystyle a\left\Vert \varphi_{3-i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{-2\beta_{i}}y_{3-i}^{1+\beta_{i}}\left(t\right),\quad t>0,\\ y_{i}\left(0\right)&\displaystyle =&\displaystyle \mathbb{E}^{i}\left[h_{i}\left(\cdot\right)\right],\quad i=1,2. \end{array} \end{aligned} $$

Putting $b:=a\min \left \{\left \Vert \varphi _{1}\right \Vert _{L^{2}\left (\mathbb {R},\mathrm {d} x\right )}^{-2\beta _{2}}, \left \Vert \varphi _{2}\right \Vert _{L^{2}\left (\mathbb {R},\mathrm {d} x\right )}^{-2\beta _{1}}\right \}$ we get the system of differential inequalities

$$\displaystyle \begin{aligned} \begin{array}{rcl} y^{\prime}_{i}\left(t\right) &\displaystyle \geq &\displaystyle b y_{3-i}^{1+\beta_{i}}\left(t\right),\quad t>0,\\ y_{i}\left(0\right)&\displaystyle =&\displaystyle \mathbb{E}^{i}\left[h_{i}\left(\cdot\right)\right],\quad i=1,2. \end{array} \end{aligned} $$

Let $\left (z_{1}\left (t\right ),z_{2}\left (t\right )\right )$ be the solution of the system of ordinary differential equations

$$\displaystyle \begin{aligned} \begin{array}{rcl} z^{\prime}_{i}\left(t\right) &\displaystyle =&\displaystyle b z_{j}^{1+\beta_{i}}\left(t\right),\quad t>0,\\ z_{i}\left(0\right)&\displaystyle =&\displaystyle \mathbb{E}^{i}\left[h_{i}\left(\cdot\right)\right],\quad i=1,2,\quad j=3-i. \end{array} \end{aligned} $$

By the Picard-Lindelöf theorem, this system with $\left (z_{1}\left (0\right ),z_{2}\left (0\right )\right )=\left (0,0\right )$ has a unique local solution $\left (w_{1}\left (t\right ),w_{2}\left (t\right )\right )\equiv \left (0,0\right )$ for all $t\in \left [0,\tau \right )$, for some $\tau \in \left (0,\infty \right ]$. In our case $\mathbb {E}^{i}\left [h_{i}\left (\cdot \right )\right ]\geq 0$. Therefore by a classical comparison theorem, $z_{1}\left (t\right ),z_{2}\left (t\right )\geq 0$ for all $t\in \left [0,\tau \right )$.

Consider the new function

$$\displaystyle \begin{aligned} E\left(t\right):=z_{1}\left(t\right)+z_{2}\left(t\right),\quad t\geq 0. \end{aligned}$$

We deal separately with the two cases in the statement of the theorem:

• 1. Case β ₁ = β ₂. Using the fact that

  $$\displaystyle \begin{aligned} x^{1+\beta_{1}}+y^{1+\beta_{1}}\geq 2^{-\beta_{1}}\left(x+y\right)^ {1+\beta_{1}},\quad x\geq0,\quad y\geq0, \end{aligned} $$

  (13)

  we get

  $$\displaystyle \begin{aligned} \begin{array}{rcl} E'\left(t\right) &\displaystyle =&\displaystyle z^{\prime}_{1}\left(t\right)+z^{\prime}_{2}\left(t\right)\\ &\displaystyle =&\displaystyle b\left( z_{1}^{1+\beta_{1}}\left(t\right)+ z_{2}^{1+\beta_{1}}\left(t\right)\right)\\ &\displaystyle \geq&\displaystyle 2^{-\beta_{1}}bE^{1+\beta_{1}}\left(t\right),\quad t>0,\\ E\left(0\right)&\displaystyle =&\displaystyle \mathbb{E}^{1}\left[h_{1}\left(\cdot\right)\right]+\mathbb{E}^{2}\left[h_{2}\left(\cdot\right)\right]. \end{array} \end{aligned} $$

  Let $I\left (t\right )$ be the solution of the ordinary differential equation

  $$\displaystyle \begin{aligned} \begin{array}{rcl} I'\left(t\right) &\displaystyle =&\displaystyle 2^{-\beta_{1}}bI^{1+\beta_{1}}\left(t\right),\quad t>0,\\ I\left(0\right)&\displaystyle =&\displaystyle \mathbb{E}^{1}\left[h_{1}\left(\cdot\right)\right]+\mathbb{E}^{2}\left[h_{2}\left(\cdot\right)\right]. \end{array} \end{aligned} $$

  Since I is a subsolution of E (see [13], Lemma 1.2.) and I explodes at time

  $$\displaystyle \begin{aligned} T^{*}=\frac{2^{\beta_{1}}} {b\beta_{1}\left(\mathbb{E}^{1}\left[h_{1}\left(\cdot\right)\right]+ \mathbb{E}^{2}\left[h_{2}\left(\cdot\right)\right]\right)^{\beta_{1}}} \in\left(0,\infty\right), \end{aligned}$$

  it follows that E explodes at some time t _E  ≤ T ^∗, and therefore, by a classical comparison theorem we get that

  $$\displaystyle \begin{aligned} & \mathbb{E}^{1}\left[w_{1}\left(t,\cdot\right)\right]=\left\Vert u_{1}\left(t,\cdot\right)\right\Vert _{L^{1}\left(\mathbb{R},\mu_{1}\right)}= \infty\quad \mbox{ or }\\ &\mathbb{E}^{2}\left[w_{2}\left(t,\cdot\right)\right]=\left\Vert u_{2} \left(t,\cdot\right)\right\Vert _{L^{1}\left(\mathbb{R},\mu_{2}\right)}=\infty \end{aligned} $$

  for all t ≥ T ^∗. Since $\left \Vert u_{i}\left (t,\cdot \right )\right \Vert _{L^{1}\left (\mathbb {R},\mu _{i}\right )}\leq \left \Vert u_{i}\left (t,\cdot \right )\right \Vert _{L^{\infty }\left (\mathbb {R},\mathrm {d} x\right )}\left \Vert \varphi _{i}\right \Vert _{L^{2}\left (\mathbb {R},\mathrm {d} x\right )}^2$ for all $t\in \left [0,\infty \right )$, i = 1, 2, we conclude that the mild solution $\left (u_{1},u_{2}\right )$ of system (2) blows up in finite time.

• 2. Case β ₁ > β ₂. Recall that for all x, y ≥ 0, δ > 0 and $p,q\in \left (1,\infty \right )$ such that p ⁻¹ + q ⁻¹ = 1 we have Young’s inequality

  $$\displaystyle \begin{aligned} xy \ \leq \ \frac{\delta^{-p}x^p}{p}+\frac{\delta^{q}y^q}{q}. \end{aligned} $$

  (14)

  From the definition of A ₀ it follows that

  $$\displaystyle \begin{aligned} z_{2}^{1+\beta_{1}}\left(t\right)\geq z_{2}^{1+\beta_{2}}\left(t\right)-A_{0},\quad \text{for all }t\geq 0. \end{aligned}$$

  In fact, it suffices to choose in (14)

  $$\displaystyle \begin{aligned} x=1,\quad y=z^{1+\beta_{2}}_{2}\left(t\right),\quad \delta=\left(\frac{1+\beta_{1}}{1+\beta_{2}}\right)^{\frac{1+\beta_{2}}{1+\beta_{1}}}\quad \mbox{and}\quad q=\frac{1+\beta_{1}}{1+\beta_{2}}. \end{aligned}$$

  Therefore we have

  $$\displaystyle \begin{aligned} E'\left(t\right) \geq b\left(z_{1}^{1+\beta_{2}}\left(t\right)+ z_{2}^{1+\beta_{2}}\left(t\right)-A_{0}\right). \end{aligned} $$

  Using again inequality (13) we conclude that

  $$\displaystyle \begin{aligned} z_{1}^{1+\beta_{2}}\left(t\right)+z_{2}^{1+\beta_{2}}\left(t\right) \geq2^{-\beta_{2}}E^{1+\beta_{2}}\left(t\right), \end{aligned}$$

  hence

  $$\displaystyle \begin{aligned} E'\left(t\right)\geq b\left(2^{-\beta_{2}}E^{1+\beta_{2}}\left(t\right)-A_{0}\right). \end{aligned}$$

  Let $I\left (t\right )$ solve the ordinary differential equation

  $$\displaystyle \begin{aligned} I'\left(t\right)&=b\left(2^{-\beta_{2}}I^{1+\beta_{2}}\left(t\right)-A_{0}\right),\quad t>0,\\ I\left(0\right)&=\mathbb{E}^{1}\left[h_{1}\left(\cdot\right)\right]+\mathbb{E}^{2}\left[h_{2}\left(\cdot\right)\right]. \end{aligned} $$

  It follows from the same comparison theorem as above that I is a subsolution of E. Using separation of variables we get, for $t\in \left (0,\infty \right )$,

  $$\displaystyle \begin{aligned} t=\int_{E\left(0\right)}^{I\left(t\right)}\frac{\mathrm{d} x}{b\left(2^{-\beta_{2}}x^{1+\beta_{2}}-A_{0}\right)}\leq \int_{E\left(0\right)}^{\infty}\frac{\mathrm{d} x}{b\left(2^{-\beta_{2}}x^{1+\beta_{2}}-A_{0}\right)}=:T^{*}. \end{aligned} $$

  (15)

  But the hypothesis (8) implies that T ^∗ < ∞. Hence (15) cannot hold for sufficiently large t, which yields that I explodes at a finite time $T^{**}\in \left (0,T^{*}\right ]$. Therefore E explodes no later than T ^∗ as well. From here we proceed as in the case β ₁ = β ₂ to conclude that the mild solution $\left (u_{1},u_{2}\right )$ of system (2) blows up in finite time also in this case.

□

The following result is an immediate consequence of the previous theorem. Recall that $E\left (0\right )=\int _{\mathbb {R}}f_{1}\,\mathrm {d}\mu _1+\int _{\mathbb {R}}f_{2}\,\mathrm {d}\mu _2$ and

$$\displaystyle \begin{aligned} A_{0}=\left(\frac{1+\beta_{2}}{1+\beta_{1}}\right)^{\frac{1+\beta_{2}}{\beta_{1}- \beta_{2}}}\frac{\beta_{1}-\beta_{2}}{1+\beta_{1}},\quad b=\min\left\{k_{1}^{2},\frac{1}{k_{2}^{2}}\right\} \min_{i\in\left\{1,2\right\}}\left\{\left\Vert \varphi_{i}\right\Vert _{L^{2}\left(\mathbb{R},\mathrm{d} x\right)}^{-2\beta_{i}}\right\} .\end{aligned}$$

Corollary 3

Under the assumptions of Theorem 2 , if β ₁ = β ₂ then the explosion time of any non-trivial positive solution of (2) is bounded above by

$$\displaystyle \begin{aligned} T^{*}=\frac{2^{\beta_{1}}} {b\beta_{1}\left(E\left(0\right)\right)^{\beta_{1}}}. \end{aligned}$$

If β ₁ > β ₂ and (8) holds, then the time of explosion of (2) is bounded above by

$$\displaystyle \begin{aligned} T^{*}=\int_{E\left(0\right)}^{\infty}\frac{\mathrm{d} x}{b\left(2^{-\beta_{2}}x^{1+\beta_{2}}-A_{0}\right)}. \end{aligned}$$

Remark

Theorem 2 and Corollary 3 remain valid when β ₂ > β ₁, with the obvious changes in the correspondent statements.