LOCAL SOLVABILITY AND BLOW-UP FOR BENJAMIN-BONA-MAHONY-BURGERS, ROSENAU-BURGERS AND KORTEWEG-DE VRIES-BENJAMIN-BONA-MAHONY EQUATIONS

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1 Electronic Journal of Differential Equations, Vol. 14 (14), No. 69, pp ISSN: URL: or ftp eje.math.txstate.eu LOCAL SOLVABILITY AND BLOW-UP FOR BENJAMIN-BONA-MAHONY-BURGERS, ROSENAU-BURGERS AND KORTEWEG-DE VRIES-BENJAMIN-BONA-MAHONY EQUATIONS MAXIM O. KORPUSOV, EGOR V. YUSHKOV Abstract. In this article some well-known problems in mathematical physics for Benjamin-Bona-Mahony-Burgers, Rosenau-Burgers an Korteweg-e Vries- Benjamin-Bona-Mahony equations are consiere. These equations escribe important processes in ifferent fiels of physics, particulary in hyro- an electroynamics. We stuy initial-bounary problems with natural physical bounary conitions. Sufficient conitions of local solvability an blow-up in finite time are obtaine. For this the methos of contraction mapping an nonlinear capacity, evelope by Galaktionov, Pokhozhaev an Mitiieri, are use. 1. Introuction The Kortweg-e Vries equation is well known in ifferent fiels of science an technology, u t + uu x + u xxx =. (1.1) Recently Pokhozaev obtaine sufficient conitions for the finite time blow-up of solutions of initial-bounary problems for KV equation [3, 4, 5, 6]. He use the powerful metho of nonlinear capacity, evelope in [18]. Note that in these papers both classical, an weak solutions of the problems were consiere. By the metho of nonlinear capacity, we stuy the following three equations which are important in ifferent physical applications such as waves on shallow water, an processes in semiconuctors with negative ifferential conuctivity: Benjamin-Bona-Mahony-Burgers equation t (u xx u) + u xx uu x =, (1.) Rosenau-Burgers equation: t (u xxxx + u) + u xx uu x =, (1.3) Mathematics Subject Classification. 35B44. Key wors an phrases. Local solvability; blow-up; Benjamin-Bona-Mahony-Burgers; Rosenau-Burgers; Korteweg-e Vries-Benjamin-Bona-Mahony. c 14 Texas State University - San Marcos. Submitte August 6, 13. Publishe March 15, 14. 1

2 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 Korteweg-e Vries-Benjamin-Bona-Mahony equation: t (u xx u) u xxx uu x =. (1.4) The literature on these equations is very extensive. Among them, we mention the classical papers [1] [7]. Also we mention our papers [1, 13, 3], sufficient conitions for blow-up in one- an multiimensional (1.), (1.3) an (1.4) equations were obtaine, but for other bounary conitions an without solving of the local solvability question. Finally, for completeness we recall that there are three main methos for stuying blow-up: The metho of nonlinear capacity, evelope by Pokhozhaev an Mitiieri [17, 18, ]; the energy metho evelope by Levine [, 11, 14, 15,, 1, 9]; an the metho, base on maximum principle, propose by Samarskii, Galaktionov, Kuryumov an Mikhailov [8, 8].. Blow-up for equation (1.) Let us consier the initial-bounary problem (1.) an t (u xx u) + u xx uu x =, x (, L), t >, (.1) u(, t) = u x (, t) =, u(x, ) = u (x), x [, L], t, (.) L (, + ). We are looking for the solution of problem (.1) (.) such that u(x)(t) C (1) ([, T ]; C () ([, L])) (.3) for some T >. By efinition, if f C (), then f C() an f() = f x () =. In this section we show the following result. Theorem.1. For any u C () ([, L]) there exists one an only one solution of problem (.1), (.) such that u(x)(t) C (1) ([, T ); C () ([, L])), either T = + or T < + an the limiting equality hols lim sup t T sup x [,L] Proof. We begin with some notation. operator G f = y y By (.3) we can rewrite (.1) as f(z) z = u(x)(t) = +. (.4) For any f C[, L] we use the Volterra t [I G ]u + u = 1 (x z)f(z) z for x [, L]. u x. (.5) The spectral raius of the Volterra operator is zero. Thus multiplying both sies (.5) by [I G ] 1, we obtain + u t + u = [G ] k u + 1 k= [G ] k u y. (.6)

3 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 3 With the new variables w = e t u, integrating (.6), we have w = F[w] u + t B 1 [w] = [G ] k w, sb[w](s), B[w] = B 1 [w] B [w], (.7) B [w] = e t k= [G ] k w y. Let us consier the following close, boune an convex subset in the Banach space C([, T ] [, L]): B R { w C([, T ] [, L]) : w sup w(x, t) R }. (t,x) [,T ] [,L] We prove that the operator F[w] : B R B R f or large enough R > an small T >. Inee, suppose that (t 1, x 1 ) an (t, x ) belong to [, L] [, T ]. Without loss of generality we suppose that t t 1. Then the following chain of inequalities hols t1 t t sb 1 [w](s, x 1 ) t 1 B 1 [w](s, x 1 ) s + t t 1 sb 1 [w](s, x ) t1 L k (k)! w + T x 1 x In the same way, for B [w] we have the estimate t1 t sb [w](s, x 1 ) t t 1 B [w](s, x 1 ) s + t t 1 1 k= B 1 [w](s, x 1 ) B 1 [w](s, x ) sb [w](s, x ) t1 L k 1 (k 1)! w. B [w](s, x 1 ) B [w](s, x ) L k+1 (k + 1)! w + T x 1 x 1 k= L k (k)! w. (.8) (.9) This yiels that the operator F is a mapping from C([, T ] [, L]) to C([, T ] [, L]). Now show that the operator is a mapping from B R to B R. It is clear that F[w] u + u + T T s B[w](s) L k (k)! w + T k= From (.1) we get that for large enough R >, an for small enough T >, T u R, L k+1 (k + 1)! w. L k (k)! R + T L k+1 (k + 1)! R R. k= (.1)

4 4 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 The result of Theorem 1 is true. Now let us emonstrate the contraction mapping F[w] on B R. It is not har to show that the following chain of inequalities hols F[w 1 ] F[w ] T B[w 1 ] B[w ] then from (.11), for any [ + T T [ + L k + (k)! + k= L k+1 ] (k + 1)! R w 1 w, L k + (k)! + L k+1 (k + 1)! R] 1 k= (.11) we have a contraction F[w] on B R. This means that the unique, local in time, solvability of the integral equation (.7) is prove. We claim that the solution w of (.7) belongs to the class C([, T ) [, L]); moreover, either T = +, or T < + an lim t T w = +. Assuming the converse: T < +, then we have From equation (.7) we get the estimate This implies that sup w < +. t [,T ] w c(t ) < + for all t [, T ]. w(x)(t 1 ) w(x)(t ) t t 1 w (x)(τ) τ c(t ) t t 1, the constant c(t ) < + oes not epen on x [, L]. Whence, for all x [, L] we can efine w(x)(t ). Furthemore, if we take supremum of both sies of the last inequality, then we obtain w(x)(t ) B[, L], B[, L] is the set of boune functions on the line segment [, L]. We shall see that w(x)(t ) C[, L]. Obviously, for all ε > there is δ(ε) > such that for all x 1, x [, L], x 1 x δ(ε)/ an we can choose t 1, t [, T ) such that imply T t 1 δ(ε)/4, T t δ(ε)/4 t 1 t T t 1 + T t δ(ε)/, t 1 t + x 1 x δ(ε), so the following chain of inequalities hols: w(x 1 )(T ) w(x )(T ) w(x 1 )(T ) w(x 1 )(t 1 ) + w(x 1 )(t 1 ) w(x )(t ) + w(x )(T ) w(x )(t ) c(t ) T t 1 + w(x 1 )(t 1 ) w(x )(t ) + c(t ) T t c(t )( ε 4 + ε + ε 4 ) = c(t )ε. (.1)

5 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 5 Therefore, we can efine w(x)(t ) C([, L]) an can exten the classical solution over the time moment T >, taking the following initial function w(x)(t) for t = T. This contraiction conclues the result of existance of a maximal solution. Finally, using so calle bootstrap metho for the integral equation (.7) uner the aitional conition u (x) C () ([, L]), we get that proof is complete. Now we shall obtain a blow-up result. w(x)(t) C (1) ([, T ); C () ([, L])). Theorem.. Suppose that for some positive λ 3, the initial function satisfies the conition J() > m k, (.13) or (L x) λ 3 [λ(λ 1) (L x) ]((L x)u (λ 1)) x, (L x) λ [λ(λ 1) (L x) ]u x > (λ 1)L λ 1 λ(λ 1) λ 1 λ (λ 1)L + λl4 λ(λ + ) λ(λ 1) + L λ λ 1 λ λ Lλ, then the classical solution of problem (.1) (.) oes not exist globally in time. Furthermore, the following estimate hols consequently, J(t) m k m = (λ 1) L λ 1/ (kj() + m) + (kj() m) exp(mkt) (kj() + m) (kj() m) exp(mkt), (.14) lim +, T b 1 t T b mk ln(kj() m kj() + m, k = L λ λ(λ 1) λ ), (.15) 1 λ (λ 1)L + λl4 1. λ(λ + ) Proof. To obtain sufficient conitions of the blow-up we use the metho of nonlinear capacity. Let us take the test function ϕ(x) = (L x) λ, λ 3 to remove bounary conitions on the right sie (x = L). Multiplying both sies of (1.) by ϕ(x) an integrating by parts over [, L], we use the following equalities (L x) λ u xx (x, t) x = λ(λ 1) (L x) λ uu x (x, t) x = λ (L x) λ u x, (L x) λ 1 u x,

6 6 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 an get the orinary ifferential equation (L x) λ [λ(λ 1) (L x) ]u x = λ(λ 1) (L x) λ u x + λ u (L x) λ 1 x. (.16) Constructing the perfect square in the right sie of (.16) an aing in the left sie the time-inepenent function, we obtain = λ (L x) λ 3 [λ(λ 1) (L x) ]((L x)u (λ 1)) x By Holer s inequality we can substitute [u(l x) (λ 1)] (L x) λ 3 x ab a 1/ λ(λ 1) a = [u(l x) (λ 1)] (L x) λ 3, b = [λ(λ 1) (L x) ] (L x) λ 3, (L x) λ 1 x. (.17) b 1/ (.18) an integrate over the line segment [, L]. Then (.18) gives the following estimate [u(l x) (λ 1)] (L x) λ 3 x J From equation (.17) we get the ODE J(t) λj b x 1 λ(λ 1) b x 1. (L x) λ 1 x, (.19) (L x) λ 3 [λ(λ 1) (L x) ]((L x)u (λ 1)) x. The final inequality (.19) can be written as k = λ = λ = L λ m = b x 1 λ(λ 1) J(t) k J m, (.) (L x) λ 1 x = (λ 1) L λ, [λ (λ 1) λ(λ 1)(L x) + (L x) 4 ](L x) λ 3 x 1 λ(λ 1) λ 1 λ (λ L 4 1)L + 1. λ(λ + ) Integrating the inequality (.), we complete the proof.

7 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 7 Note that the Theorem. is prove for any fixe L (, + ). But we shoul stress that the time moment T > is the function of L > such that the following situation is possible: T + as L +. Theorem.3. Suppose that the initial ata satisfies the inequalities or, which is the same, J() > 1, λ > 1, (.1) (u λ)e λx x, u (x)e λx x >. Then the global in time solution of the problem (.1), (.) oes not exist in the half-space (L = ). Furthermore, J(t) 1 + C exp(λ t/(λ 1)) 1 C exp(λ t/(λ 1)), (.) lim +, T b λ 1 t T b λ ln C, C = J() 1 J() + 1. (.3) Proof. To remove conitions for L = + we use the test function ϕ(x) = exp( λx), λ > 1. Multiplying both sies of (1.) by ϕ(x), we integrate by parts over [, ). By the bounary conitions we have the following equalities e λx u xx (x, t) x = λ e λx u(x, t) x, e λx uu x (x, t) x = λ So we can rewrite the equation (.1) as e λx [λ 1]u x = λ e λx u x + λ e λx u (x, t) x. e λx u x. (.4) As above, we make the perfect square in the right sie of (.4) an a the timeinepenent function in the left sie, we obtain e λx [λ 1](u λ) x = λ It is not har to prove the inequality (u λ)e λx x = 1 λ Thus from (.5) we get the ODE J(t) (u λ) e λx x λ3 (u λ) e λx x (u λ) e λx x. e λx x e λx x. (.5) (.6) λ (λ 1) J λ (λ 1), (.7)

8 8 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 (u λ)e λx x. Integrating inequality (.7), we obtain the state result. 3. Blow-up for the Rosenau-Burgers equation Let us consier (1.3) with the initial-bounary conitions t (u xxxx + u) + u xx uu x =, x (, L), t >, (3.1) u(, t) = u x (, t) = u xx (, t) = u xxx (, t) =, u(x, ) = u (x), (3.) L (, + ). Assume that there is such T >, that classical solution u of the problem (3.1) (3.) exists an satisfies u(x)(t) C (1) ([, T ]; C (4) ([, L])). (3.3) By efinition, if f C (4), then f C(4) an f() = f x () = f xx () = f xxx () =. In this section we prove the following result. Theorem 3.1. For any initial ata u C (4) ([, L]) there exists only one solution u of problem (3.1), (3.) such that u(x)(t) C (1) ([, T ); C (4) ([, L])), for some T > such that either T = + or T < + an lim sup t T sup x [,L] u(x)(t) = +. (3.4) The proof of the above theorem is the same as that for the Theorem.1. So we omit it. Let us prove now a blow-up result. Theorem 3.. Suppose that there exists such λ 7 that initial function satisfies the inequality J() > m k, (3.5) (L x) λ 5 [λ(λ 1)(λ )(λ 3) + (L x) 4 ]((L x)u (λ 1)) x. then the the classical solution of (3.1) oes not exist globally in time. Moreover, for m an k from (3.) the following estimate hols an, consequently, k = L6 λ J(t) m k (kj() + m) + (kj() m) exp(mkt) (kj() + m) (kj() m) exp(mkt), (3.6) lim +, (3.7) t T b T b 1 mk ln(kj() m kj() + m ), m = (λ 1) L λ, λ(λ 1) (λ ) (λ 3) + (λ 1)(λ 3)L 4 L 8 + (λ 6) λ(λ + ) 1

9 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 9 Proof. Multiplying both sies (3.1) by ϕ(x) = (L x) λ, we integrate by parts. By the bounary conitions, we have (L x) λ u xxxx (x, t) x = λ(λ 1)(λ )(λ 3) (L x) λ 4 u(x, t) x; By this equality we can rewrite (1.3) as the integro-ifferential expression (L x) λ 4 [λ(λ 1)(λ )(λ 3) + (L x) 4 ]u x = λ(λ 1) (L x) λ u x + λ u (L x) λ 1 x. (3.8) As before, in the right-han sie of (3.8) we complete the square an in the left-han sie a the time-inepenent function: = λ (L x) λ 5 [λ(λ 1)(λ )(λ 3) + (L x) 4 ]((L x)u (λ 1)) x Substituting [u(l x) (λ 1)] (L x) λ 3 x λ(λ 1) a = [u(l x) (λ 1)] (L x) λ 3, b = [λ(λ 1)(λ )(λ 3) (L x) 4 ] (L x) λ 7 in (.18) an integrating over the segment [, L], we obtain [u(l x) (λ 1)] (L x) λ 3 x J b x 1 (L x) λ 1 x. (3.9) = J λ (λ 1) (λ ) (λ 3) L λ 6 + λ(λ 1)(λ 3)L λ + Lλ+ 1. (λ 6) λ + (3.1) Finally, from (3.9) we obtain the orinary ifferential inequality m = (λ 1) L λ, J(t) k J m, (3.11) k = L6 λ λ(λ 1) (λ ) (λ 3) + (λ 1)(λ 3)L 4 L 8 + 1, (λ 6) λ(λ + ) (L x) λ 5 [λ(λ 1)(λ )(λ 3) + (L x) 4 ]((L x)u (λ 1)) x. Integrating (3.11), we complete the proof. As for equation (1.), we can obtain results for the Rosenau-Burgers in the unboune omain. For problem (3.1)-(3.) on the half-line [, + ) we shall prove the following result. Theorem 3.3. Suppose that the initial ata satisfies the inequality u (x)e λx x >,

10 1 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 then the classical solution of the problem for Rosenau-Burgers on the half line oes not exist globally in time. Furthermore, there exists the lower estimate J(t) 1 + C exp(λ t/(λ 4 + 1)) 1 C exp(λ t/(λ 4 + 1)), an, consequently, (u λ)e λx x, (3.1) lim +, T b λ4 + 1 t T b λ ln C, C = J() 1 J() + 1. (3.13) Proof. The proof is similar to the Theorem.3; so we present on the main points. First let us multiply both sies of (1.3) by ϕ(x) = exp( λx), with λ > 1, an integrate by parts over [, L): e λx [λ 4 + 1]u x = λ e λx u x + λ e λx u x. (3.14) Then extracting perfect square in the sie part (3.14) an aing a time-inepene function to the left-han sie, we obtain the equality e λx [λ 4 + 1](u λ) x = λ By (.6) we can rewrite (3.15) as J(t) (u λ) e λx x λ3 e λx x. (3.15) λ (λ 4 + 1) J λ (λ 4 + 1), (3.16) (u λ)e λx x. Integrating (3.16), we obtain the result of the theorem. 4. Blow-up for equation (1.4) It is reaily seen that the metho of nonlinear capacity is usable for Sobolev type equations also with aitional terms like higher-orer erivatives u (n). As an example, we consier the problem for the equation, which escribes processes in cristalline semiconuctors an has the thir-orer erivative: (u xx u) t + u xxx uu x =, t >, x (, L), L >, (4.1) u(, t) = u x (, t) = u xx (, t) =, u(x, ) = u (x), x [, L], t. (4.) The physical moel for problem (4.1) (4.) is presente in [31]. Let us prove the solvability result. Theorem 4.1. For any function ũ (x) such that { u (x), for x ; ũ (x) = ũ (x) C (4) (, L],, for x <, there exists only one solution u of (4.1), (4.) such that u(x)(t) C (1) ([, T ); C (3) ([, L])),

11 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 11 for some T >. By efinition, if f C (3), then f C(3) an f() = f x () = f xx () =. Moreover, it can be prove that either T = + or T < + an the following limiting equality hols sup sup t T x [,L] u(x)(t) = +. Remark 4.. By the bounary conitions (4.) an natural matching conition of initial an bounary ata it is easily shown that ũ (x) C (3) (, L], however, larger smoothness is not inispensable. In aition, it is clear that necessary an sufficient conitions for ũ (x) C (4) (, L] are the following u (x) C (4) ([, L]) an (u ) xxx ( + ) = (u ) xxxx ( + ) =. Proof of Theorem 4.1. As above for equation (1.), taking into account the bounary conitions, integrating with respect to x an inverting the operator we obtain the equation (I G ), G f = k= u t + u + x = Q[u](x, t) [G ] k u y + 1 we use that u y y = u(x), an, thus, [G ] k u x = k= y y k= [G ] k u y. f(z) z, [G ] k u y, (4.3) From the efinition of operator Q[u](x, t) it follows that Q[u](, t) =, moreover, u(, t) =. Then we can continue the function Q[u](x, t) by zero for x <, furthermore, this function Q[u](x, t) is smooth function with respect to the first input. Corresponing to this continuation we efine Now consier the ifferential problem f(x, t) Q[u](x, t) an ũ(x, t). v t + v x = f(x, t), v(x, ) = v (x), v(x, t) ũ(x, t). (4.4) The solution of (4.3) we can be written as u(x, t) = ũ (x t) + t f(x t + τ, τ) τ, x [, L], t [, T ], (4.5) L > is fixe, an T > is small enough. To prove the local solvability of the integral equation (4.5) we use the contraction mapping metho. In the notation of the previous section we can rewrite the integral equation (4.5) in the abstract form u(x, t) = F[u](x, t) ũ (x t) + t Q[u](x t + τ, τ) τ. (4.6) As before, it can easily be checke that, if u (x) C (3) ([, L]), then F[ ] : C([, T ] [, L]) C([, T ] [, L]).

12 1 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 Let us prove now that F[ ] is an operator between B R an B R for large enough R > an small T >. Inee, we can fix R > such large that u R. On the other han, we have the inequality u u + T k= From (4.7), for small enough T >, (L ) k+1 [ 1 u + (k + 1)! u ], L = max{l, L T }. (4.7) T k= (L ) k+1 R [R + (k + 1)! ] R, an we obtain our result. As in the secon section it is easily shown that F is contraction mapping on a space B R, if we change L to L. Thus, we complete the proof of local in time solvability of the integral equation (4.5). Now we can continue the solution in time. Assume that u (x) C (3) ([, L]), then from the necessary matching conition of initial an bounary ata we obtain that ũ (x) C (4) ((, L]). Moreover, it can easily be checke that u(x)(t) C (3) ([, T ] [, L]). Let us enote t w(x)(t) ũ (x t) + Q[u](x t + τ, τ) τ, then uner the conition T < +, the following inequality hols sup u < +. t [,T ] By a stanar way we obtain w(x)(t ) C([, L]). At the same time we can write w(x)(t ) = ũ (x T ) + T Q[u](x T + τ, τ) τ C (1) ([, L]). Therefore, in (4.3), taking u(x)(t) for t = T as initial ata, we can continue the solution over the time moment T. This contraiction conclues the proof of existence of the maximum solution. Using the bootstrap metho, it is possible to show that, if ũ (x) C (4) (, L], then the solution of (4.5) belong to the class C (1) ([, T ); C (3) ([, L])). As above, to obtain sufficient conitions of blow-up we use the metho of nonlinear capacity for the power test function ϕ(x) = (L x) 5. Suppose that the solution u(x, t) of problem (1.4), (4.) is classical: u(x)(t) C (1) ([, T ]; C (3) ([; L])) for some T >. Multiplying both sies of (1.4) by the test function ϕ(x) an integrating by parts, we obtain the equality: (L x) 3 [ (L x) ]u x = 5 u (L x) 4 x+6 u(l x) x. (4.8)

13 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 13 As before, let us make in the right-hna sie (.16) the perfect square an in the left a the time-inepenent function (L x)[ (L x) ]((L x) u + 1) x = 5 The following inequalities can be easily prove: (L x)[ (L x) ]((L x) u + 1) x = 5 k (L x) [ (L x) ] x ((L x) u + 1) x, [(L x) u + 1] x 36L. ((L x) u + 1) x (4.9) (4.1) k = 16L3 16L5 + L From equation (4.9) an estimate (4.1) we obtain the orinary ifferential inequality J(t) k J 36L, (4.11) (L x)[ (L x) ]((L x) u + 1) x. Integrating (4.11), we obtain the following result. Theorem 4.3. Let the initial function satisfies the conition J() = (L x)[ (L x) ]((L x) u (x) + 1) x > 36L k (4.1) or (L x) 3 [ (L x) ]u (x) x > 1L 4 3 L 5 + L4 7 1L + 3L 4, then there is no global in time classical solutions of (1.4). Moreover, the following lower boun hols J(t) 6 1L 1 + C exp(1 1Lkt) k 1 C exp(1 1Lkt), (4.13) an, consequently, lim t Tb +, 1 T T b 1 ln C, 1Lk 16L3 16L5 3 5 C = kj() 6 1L kj() + 6 1L, k = + L Using the nonlinearity capacity metho with the test function ϕ(x) = exp( λx), λ > 1, we obtain the blow-up result on the halfline. Multiplying both sies of (1.4) equation by ϕ(x) an integrating by parts, we obtain the equality: e λx [λ 1]u x = λ 3 e λx u x + λ e λx u x. (4.14)

14 14 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 Similarly to the boune omain, forming perfect square an aing time-inepenent function, we obtain that e λx [λ 1](u + λ ) x = λ (u + λ ) e λx x λ5 It is not har to prove the following chain of inequalities: (u + λ )e λx x + (u + λ ) e λx x Thus, we can rewrite (4.15) as J(t) = 1 λ (u + λ ) e λx x. e λx x e λx x. (4.15) (4.16) λ (λ 1) J λ 4 (λ 1), (4.17) (u + λ )e λx x. After integrating (4.17), we can formulate the following statement. Theorem 4.4. Suppose that the initial ata satisfies the conition J() = (u (x) + λ )e λx x > 1, λ > 1. (4.18) Then the classical solution of (1.4), (4.) oes not exist globally in time. Furthermore, the following inequality hols consequently, J(t) 1 + C exp(λ3 t/(λ 1)) 1 C exp(λ 3 t/(λ 1)), (4.19) lim +, T b λ 1 t T b λ 3 ln C, C = J() λ J() + λ. (4.) Conclusion. This article shows that the nonlinear capacity metho allow us to stuy not only stanar Kortweg-e Vries equations, but also other problems of moern mathematical physics: problems for Benjamin-Bona-Mahony-Burgers, Rosenau-Burgers an Korteweg-e Vries-Benjamin-Bona-Mahony equations. Moreover, it is possible to unite the proof of blow-up in problems with nonclassical bounary conitions an the existence of correct blow-up solutions. This research was supporte by RFBR (grant a, ofi-m- 11) an by grant from the Presient of the Russian Feeration for young Russian scientists MD References [1] Albert, J. P.; On the ecay of solutions of the generalize Benjamin-Bona-Mahony equation, J. Math. Anal. Appl., 141, No., (1989). [] Al shin, A. B.; Korpusov, M. O.; Sveshnikov, A. G.; Blow-up in nonlinear Sobolev type equations. De Gruyter, Series: De Gruyter Series in Nonlinear Analysis an Applications , 448. [3] Avrin, J. D.; Golstein, J. A.; Global existence for the Benjamin-Bona-Mahony equation in arbitrary imensions, Nonlinear Anal., 9, No. 8, (1985).

15 EJDE-14/69 LOCAL SOLVABILITY AND BLOW-UP 15 [4] Benjamin, T. B.; Bona, J. L.; Mahony, J. J.; Moel equations for long waves in nonlinear ispersive systems, Philos. Trans. Royal Soc. Lonon, Ser. A, 7, (197). [5] Bisognin, E.; Bisognin, V.; Charao, C. R.; Pazoto, A.F.; Asymptotic expansion for a issipative Benjamin-Bona-Mahony equation with perioic coefficients, Port. Math. (N.S.) 6 (3), no. 4, [6] Biler, P.; Long-time behavior of the generalize Benjamin-Bona-Mahony equation in two space imensions, Differ. Integral Equations, 5, No. 4, (199). [7] Chen Yu.; Remark on the global existence for the generalize Benjamin-Bona-Mahony equations in arbitrary imension, Appl. Anal., 3, Nos. 1 3, 1 15 (1988). [8] Galaktionov, V. A., Pohozaev, S. I.; Thir-orer nonlinear ispersive equations: Shocks, rarefaction, an blowup waves, Com. Math. Math. Phys, V. 48, Num. 1, [9] Hagen, T.; Turi, J.; On a class of nonlinear BBM-like equations, Comput. Appl. Math., 17, No., (1998). [1] Hayashi, N.; Kaikina, E. I.; Naumkin, P.I.; Shishmarev, I. A.; Asymptotics for Dissipative Nonlinear Equations, Springer-Verlag, New York (6). [11] Kalantarov, V. K.; Layzhenskaya, O. A.; Forming of collapses in a quasi-linear equations of parabolic an hyperbolic types (in Russian), Notes of LOMI, 1977, 69, [1] Korpusov, M. O.; On the blow-up of solutions of the Benjamin-Bona-Mahony-Burgers an Rosenau-Burgers equations, Nonlinear Analysis 75 (1) [13] Korpusov, M. O.; Blowup of solutions of the three-imensional Rosenau Burgers equation, Teoret. Mat. Fiz., 1, V. 17, N 3, pp [14] Levine, H. A.; Toorova, G.; Blow up of solutions of the Cauchy problem for a wave equation with nonlinear amping an source terms an positive initial energy in four space-time imensions, Proc. Amer. Math. Soc. 3, 19, [15] Levine, H. A.; Instability an nonexistence of global solutions to nonlinear wave equations of the form P u tt = Au + F (u). Transactions of the American mathematical society. 1974, 1 1. [16] Mei, M.; Liu, L.; A better asymptotic profile of Rosenau-Burgers equation, Appl. Math. Comput. 131 (), no. 1, [17] Mitiieri, E. L.; Galaktionov, V. A.; Pohozaev, S. I.; On global solutions an blow-up for Kuramoto-Sivashinsky-type moels, an well-pose Burnett equations. Nonlinear Anal., Theory Methos Appl. 9, 7(8), [18] Mitiieri, E.; Pokhozhaev, S. I.; A priori estimates an the absence of solutions of nonlinear partial ifferential equations an inequalities, (in Russian) Tr. Mat. Inst. Steklova, 34 (1), [19] Park, M. A.; On the Rosenau equation in multiimensional space, J. Nonlin. Anal., 1, No. 1, (1993). [] Pucci, P.; Serrin, J.; Some new results on global nonexistence for abstract evolution equation with positive initial energy, Topological Methos in Nonlinear Analys, Journal of J. Schauer Center for Nonlinear Stuies. 1997, 1, [1] Pucci, P.; Serrin, J.; Global nonexistence for abstract evolution equations with positive initial energy, J. Diff. Equations. 1998, 15, [] Pohozaev, S. I.; Critical nonlinearities in partial ifferential equations. Milan J. Math. 9, 77:1, [3] Pokhozhaev, S. I.; On the nonexistence of global solutions for some initial-bounary value problems for the Korteweg-e Vries equation, Differ. Equ., 47:4 (11), [4] Pokhozhaev, S. I.; Riemann quasi-invariants, Sb. Math., :6 (11), [5] Pokhozhaev, S. I.; Weighte Ientities for the Solutions of Generalize Korteweg-e Vries Equations, Math. Notes, 89:3 (11), [6] Pokhozhaev, S. I.; The absence of global solutions of KV equation, Journal of Mathematical Sciences, Vol. 19, No. 1, April, 13, [7] Rosenau, Ph.; Extening hyroynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 4(1): (1989). [8] Samarsky, A. A.; Galaktionov, V. A.; Kuryumov, S. P.; Mikhailov, A. P.; Sharpening Regimes in Problems for Quasilinear Parabolic Equations (in Russian), Nauka, Moscow (1987). [9] Straughan, B.; Further global nonexistence theorems for abstract nonlinear wave equations. Proc. Amer. Math. Soc., 1975,,

16 16 M. O. KORPUSOV, E. V. YUSHKOV EJDE-14/69 [3] Yushkov, E. V.; Blowup of solutions of a Korteweg-e Vries-type equation, Teoret. Mat. Fiz., 1, V. 17, N 1, pp [31] Yushkov, E. V.; Traveling-wave solution to a nonlinear equation in semiconuctors with strong spatial ispersion, Zh. Vychisl. Mat. Mat. Fiz., 8, V. 48, N 5, pp Maxim O. Korpusov Chair of Mathematics, Physics Faculty, M. V. Lomonosov Moscow State University, Moscow, Russia aress: korpusov@gmail.com Egor V. Yushkov Chair of Mathematics, Physics Faculty, M. V. Lomonosov Moscow State University, Moscow, Russia aress: yushkov.msu@mail.ru