Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

Junping Zhao

College of Science, Xi’an University of Architecture & Technology, Xi’an 710055, China Reprint requests to J. P. Z.; E-mail:junpingzhao@yeah.net

Z. Naturforsch.67a,479 – 482 (2012) / DOI: 10.5560/ZNA.2012-0048 Received February 2, 2012 / revised May 4, 2012

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut=div(a(u)b(x)∇u) +f(x,u,|∇u|²,t)under nonlinear boundary condition∂u/∂n+g(u) =0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up timeT^∗ may depend ona(u), while being independent ofg(u)and f.

Key words:Reaction-Diffusion Equation; Gradient Term; Nonlinear Boundary Conditions;

Maximum Principles; Blow-Up of Solutions.

Mathematics Subject Classification 2000:35K20, 35K55, 35K65

1. Introduction and Main Results

There are many works on the existence and non- existence of global solutions to the parabolic equa- tions and systems with various boundary conditions (see [1–7] and the references therein). In this paper, we consider the following reaction-diffusion equation with a gradient term under nonlinear boundary condi- tion:

u_t=div(a(u)b(x)∇u)

+f(x,u,|∇u|²,t), in Ω×(0,T),

∂u

∂n+g(u) =0, on ∂ Ω×(0,T), u(x,0) =u₀(x)>0, in ¯Ω,

(1)

whereΩ is a smooth bounded domain ofR^N,N≥2, n denotes the outer unit normal on the boundary,a∈ C²(R⁺),b∈C¹(Ω¯),f∈C¹(Ω¯×R×R⁺×[0,T]),g∈ C²(R⁺), f>0, andg>0.

Models such as (1) are worthy to study not only because of the importance to the qualitative theory of nonlinear partial differential equations but also be- cause of possible applications in the field of mechan- ics, physics, and biology (gas flow in porous media, semiconductor, spread of biological populations, etc.);

see [3–13] and the references therein.

Some special cases of (1) have been discussed by many authors. The blow-up phenomenon of (1) with linear boundary conditions have been extensively stud- ied over the past decades. In [14,15], Souplet dis- cussed the following equations with Dirichlet bound- ary conditions

u_t=∆u+f(x,u,|∇u|^p,t), in Ω×(0,T), u(x,t) =0, on ∂ Ω×(0,T),

u(x,0) =u₀(x)>0, in ¯Ω,

(2)

where re-scaling arguments, the self-similar subsolu- tion method, and the energy functional method are in- troduced to investigate the sufficient conditions and the qualitative properties for blow-up.

In [16], Ding studied the semilinear reaction diffu- sion equation

u_t=∆u+f(x,u,|∇u|²,t) (3) with mixed linear boundary conditions. By using the Hopf maximum principle, the author obtained the nonexistence theorems of the global solutions and gave the bound of blow-up time.

In [17], Hu and Yin considered the heat equation ut=∆uin an unbounded domainΩ⊂R^Nwith Dirich- let conditionu(x,t) =0 on partial boundary ofΩ and nonlinear condition∂u/∂n=u^pon the other part of

c

2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

480 J. P. Zhao·Blow-Up of Solutions for a Class of Reaction-Diffusion Equations the boundary, where p>1. It has been shown that the

nonnegative solutions blow up under certain conditions depending on the exponentp.

Recently, Ding and Guo [18] studied the blow- up and global solutions for the following reaction- diffusion equation with a gradient term and nonlinear boundary condition:

(g(u))_t=∆u+f(x,u,|∇u|²,t), in Ω×(0,T),

∂u

∂n+g(u) =0, on ∂ Ω×(0,T), u(x,0) =u₀(x)>0, in ¯Ω.

By constructing suitable auxiliary functions and using maximum principles, they proved the sufficient condi- tions for the existence of a blow-up solution, an up- per bound for the blow-up time, an upper estimate of the blow-up rate, the sufficient conditions for the exis- tence of the global solution, and an upper estimate of the global solution. However, as in [17], the blow-up timeT^∗still depends ong(u).

In this paper, we extend the result in [16,18] to the quasilinear reaction-diffusion equation with a gradi- ent term under nonlinear boundary condition. By con- structing a new auxiliary function, we indicate that the blow-up timeT^∗may depend ona(u), while be inde- pendent ofg(u)andf. Hence, our results in the present paper are also better than those obtained in [19], in whichT^∗≤ ¹

β R+∞

M0

a(s)

f(s)depend on botha(u)andf. It is known that, by the classical theory for parabolic equations and Hopf’s maximum principle [20], the problem (1) has a unique local solution, andu>0 since

f>0.

Our main result of this paper reads as follows:

Theorem 1. Let u be a C³(Ω ×(0,T))∩C²(Ω¯ × [0,T))solution of (1). Suppose the following:

(1) For(x,s,d,t)∈Ω×R⁺×R⁺×R⁺, a(s)>0, a⁰(s)≥0, a(s)−a⁰(s)≥0,

b(s)>0, (4)

g(s)>0, g⁰(s)≥0, ag≥(ag)⁰, (5) f_t⁰(x,s,d,t)≥0,

f_d⁰(x,s,d,t)≥ −a(s)b(x)/2, f_s⁰(x,s,d,t)−f(x,s,d,t)≥0.

(6)

(2) The constant β=min

Ω¯

a(u₀)e^−u0[div(a(u₀)b(x)∇u₀) +f(x,u0,|∇u₀|²,0)] >0.

(7)

(3) The integration Z _+∞

M₀

a(s)

e^s−1ds<+∞, where M₀=u₀(x₀) =max

Ω¯

u₀(x)for some u₀∈Ω¯. Then, u(x,t)must blow up in finite time T^∗and

T₂^∗≤ 1 β

Z +∞

M0

a(s)

e^s−1ds, (8)

as well as

u(x,t)≤G⁻¹(β(T−t)), (9) where

G(z) = Z +∞

z

a(s)

e^s−1ds, z>0. (10) 2. Proof of the Main Result

In this section, we will give a detailed proof of our result. We denoteq=|∇u|²below for convenience. We have the following Lemma:

Lemma 1. Let the conditions of Theorem1hold, then

u_t≥0 (11)

for all(x,t)∈Ω×(0,T).

Proof. Set

w=au_t (12)

then we have

∇w=a⁰u_t∇u+a∇u_t, (13) 4w=a⁰⁰u_tq+2a⁰∇u·∇u_t+a⁰u_t∆u+a∆u_t, (14) and

w_t=a⁰u²_t+a(ab∆u+a⁰bq+a∇b·∇u+f)_t

=a⁰u²_t+a²b∆u_t+aa⁰bu_t∆u+aa⁰⁰bu_tq +2aa⁰b∇u·∇u_t+aa⁰u_t∇b·∇u

+a²∇b·∇ut+a f_u⁰ut+2a f_q⁰∇u·∇ut+a f_t⁰. (15)

Combining (14) with (15), we have

w_t−ab4w=a⁰u²_t+aa⁰u_t∇b·∇u+a²∇b·∇u_t +a f_u⁰u_t+2a f_q⁰∇u·∇u_t+a f_t⁰. (16)

J. P. Zhao·Blow-Up of Solutions for a Class of Reaction-Diffusion Equations 481

From (12), we obtain

∇u_t=−1

a²a⁰∇uw+1 a∇w. With (15) and (16), we have

w_t−ab4w− a∇b+2f_q⁰∇u

·∇w +

2

aa⁰f_q⁰q−f_u⁰

w=a⁰u²_t+a f_t⁰. (17) Assumptions (4) and (5) guarantee that the right side in equality (17) is nonnegative. Therefore, we have

w_t−ab4w−(a∇b+2f_q⁰∇u)·∇w +

2

aa⁰f_q⁰q−f_u⁰

w≥0, in Ω×(0,T),

∂w

∂n +(ag)⁰

a w=0, on ∂ Ω×(0,T), w(x,0) =w0(x)≥0, in ¯Ω.

(18)

Combining (18) and (6) with maximum principles, it follows thatwcannot attain its minimum inΩ×(0,T).

Therefore, we have in ¯Ω ×[0,T),w(x,t)≥0, then ut≥0.

Proof of Theorems1. Set

v=−a(u)u_t+βe^u=−w+βe^u. Then we find

∇v=−∇w+βe^u∇u, (19) 4v=−4w+βe^uq+βe^u4u, (20) and

vt=−w_t+βe^uut. (21) Combining (19) – (21), we have

v_t−ab4v

=−w_t+ab4w+βe^uu_t−abβe^uq−abβe^u4u

=−(a∇b+2f_q⁰∇u)·∇w+ 2

aa⁰f_q⁰q−f_u⁰

w−a⁰u²_t

−a f_t⁰+βe^uu_t−abβe^uq−abβe^u4u

= (a∇b+2f_q⁰∇u)·(∇v−βe^u4u)−2

aa⁰f_q⁰q(v−βe^u) +f_u⁰v−βe^uf_u⁰−a⁰u²_t −a f_t⁰+βe^uut

−abβe^uq−abβe^u4u.

(22)

Therefore,

v_t−ab4v−(a∇b+2f_q⁰∇u)·∇v+

2

aa⁰f_q⁰q−f_u⁰

v

=−aβe^u∇b·∇u−2βe^uf_q⁰q+2a⁰ a βe^uf_q⁰q

−a⁰u²_t−a f_t⁰+βe^uu_t−abβe^uq−abβe^u4u. (23)

From (1), we obtain

ab4u=u_t−a⁰bq−a∇b·∇u−f. Therefore, we have

v_t−ab4v−(a∇b+2f_q⁰∇u)·∇v+

2

aa⁰f_q⁰q−f_u⁰

v

=−a⁰u²_t −2βe^uqa−a⁰ a

f_q⁰+ab

2

−a f_t⁰ +aβe^u(f−f_u⁰).

(24)

Assumptions (4) – (6) guarantee that the right side in equality (24) is nonpostive, i.e.,

v_t−ab4v−(a∇b+2f_q⁰∇u)·∇v +

2

aa⁰f_q⁰q−f_u⁰

v≤0. (25)

It follows from (7) that maxΩ¯

v(x,0) =max

Ω¯

−a(u₀)[div(a(u₀)b(x)∇u₀) +f(x,u0,q₀,0)] +βe^u0 =0.

(26)

On∂ Ω×(0,T), we have

∂v

∂n−(ag)⁰ a v=−

g−(ag)⁰ a

βe^u≤0. (27) Combining (25) – (27) and Hopf’s maximum principle, we have in ¯Ω×[0,T),

v≤0 and

a(u)

e^u u_t≥β. (28)

Without loss of generality, we let u0(x₀) =M0, inte- grating (28) fromM₀tou(x₀,t), we have

1 β

Z u(x₀,t) M₀

a(s) e^s ds≥t. Since

1 β

Z +∞

M₀

a(s)

e^s ds<+∞,

482 J. P. Zhao·Blow-Up of Solutions for a Class of Reaction-Diffusion Equations thenumust blow-up within finite timet=T₂^∗, and

T₂^∗≤ 1 β

Z +∞

M₀

a(s) e^s ds.

By the same argument as in [19], we can obtain (9), the upper estimates of blow-up rate.

3. Application

In this section, we will give a example to which the main theorem obtained in this paper may be applied.

Example 1. Let u be a C³(Ω×(0,T))∩C²(Ω¯ × [0,T))solution of the problem

u_t=div

e^u/2

1−∑³

i=1

x²_i

∇u

+7exp

2u+2q+2t−8

3

∑

i=1

x²_i

, in Ω×(0,T),

∂u

∂n+e^u/2=0, on ∂ Ω×(0,T), u(x,0) =u₀(x) =1−∑³

i=1

x²_i, in ¯Ω, where Ω =

x= (x₁,x2,x3)

3

∑

i=1

x_i²< ¹₂

, 0<T <

+∞. Then

a(u) =g(u) =e^u/2, b(x) =u0(x) =1−

3 i=1

∑

x²_i,

f(x,u,q,t) =7 exp u

2+2q+2t−8

3

∑

i=1

x²_i

.

It is easy to check that (4) – (6) hold. By (7), we find β=min

Ω¯

a(u₀)e^−u0

a(u₀)b(x)∆u₀+a⁰(u₀)b(x)|∇u₀|² +a(u₀)∇b(x)·∇u₀+f(x,u₀,q₀,0)

=min

Ω¯

b(x)∆u₀+1

2b(x)|∇u₀|²+∇b(x)·∇u₀ +7 exp

2q₀−8

3 i=1

∑

x²_i

=min

Ω¯

12

3

∑

i=1

x²_i −2 ₃

∑

i=1

x²_i 2

+1

= min

0≤s≤¹₂

{12s−2s²+1}=1.

It follows from Theorem1thatu(x,t)must blow-up in finite timeT₂^∗and

T₂^∗≤ 1 β

Z +∞

M₀

a(s) e^s−1ds=

Z +∞

1

e^s/2 e^s−1ds

=ln √

e+1

√e−1

.

Acknowledgements

This work was supported by the Scientific Research Program of the Shannxi Higher Education Institutions (Grant No. 12JK0868).

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