On the perturbation propagation in the initial-boundary value problem for quasilinear first order equations

Publicacions Ma emá iques, Vol 37 (1993), 209-223 . A bs ac ON THE PERTURBATION PROPAGATION IN THE INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR FIRST ORDER EQUATIONS in he domain wi h he condi ions Yu . G . RYKOV The pape deals wi h ini ial-bounda y alue p oblem o gene - alized solu ions o single quasilinea nonau onomous conse a ion law . Fo he case so-called "p ocesses wi h agg a a ion" he local- iza ion p ope y and inne boundedness a e s udied . Also in case when bounda y unc ion ends o ze o as => +oo he localiza ion e ec is ega ded . 1 . In oduc ion This pape s udies gene alized solu ions o he equa ions in he o m (1 .1)  Lu - u + [A( , x, u)],, + B( , x, u) = H( , x) Q={( ,x) : e(0,T), 0<T<+oc, xER + } u(0, x) = 0,  u( , 0) = ul ( ) . such ha He e  A( , x, u)  and  B( , x, u)  a e  con inuous  unc ions A( , x, 0)  =  B( , x, 0)  =  0 ; B( , x, u) is mono onically inc easing in u ; A( , x, u) is con inuosly di e en iable wi h espec o u, x ; A u  > 0 ; A( , 0, u)  # 0 ; A x ( , x, u) + B( , x, u)  >_ 0 ; H( , x) is a measu able unc ion bounded o bounded ; ul E Cl ([0, T», u, > 0 . The de ini ion o gene alized solu ion and p oo s o he exis en e and uniqueness heo ems can be ound in [3], [4], [7], [8] o [10] . 210  Y . G . RYKOV In Sec ion 2 he de ini ion o gene alized solu ion and compa ison he- o em a e gi en . In Sec ion 3 we deal wi h he case when he e exis s T< +oo such ha ui (T - 0) = +oo . Acco ding o he e minology o [1], [5] i co esponds o he so-called "p ocesses wi h agg a a ion" . De ini ion 1 .1 . One says ha localiza ion in he p oblem (1 .1), (1 .2) occu s i he e exis s X > 0 such ha u( , x) - 0 o x >_ X, 0 <_ <_ T . One says ha localiza ion does no occu i o e e y su icien ly la ge x, k > 0 he e exis s * > 0 such ha u( * , x,,) :,~ 0 . In he pape [1] au onomous equa ions wi h powe nonlinea i ies and ze o lowe o de e m we e s udied . The e necessa y and su icien cón- di ions o he occu ence o localiza ion and o inne boundedness o solu ions we e ob ained . In Sec ion 3 we shall s udy such ques ions o a bi a y nonlinea i ies and in he nonau onomous case . Sec ion 4 is de o ed o localiza ion in he case when ui ( ) is de ined o e e y E [0, +oo) and may end o ze o as => +oo . Some supplemen a y esul s on he localiza ion a e gi en in Sec ion 5 o he equa ion (1 .3)  u + (T - )p(u'''),, + (T - )qu , = 0,  ( , x) E (0, T) x R + . The e a e ce ain peculia i ies o he on beha io in his case . 2 . The de ini ion o gene alized solu ion . A compa ison heo em Now, le us in oduce he no ion o gene alized solu ion . De ini ion 2 .1 . A measu able unc ion u( , x) bounded o bounded is called a gene alized solu ion (abb e ia ion : g .s .) o he p oblem (1 .1), (1 .2) in Q i : 1) o e e y w ( , x) > 0, w E Có (Q) he inequali y A {lu( , x) - sjw + sign(u( , x) - s) [A( , x, u( , x)) - A( , x, s)]w x - - sign(u( , x) - s) [A x ( , x, s) + B( , x, u( , x)) - H( , x)]w} d dx > 0 holds, whe e s = cons is a bi a y ; 2) he e exis s a se El C [0, T], mes El = 0, such ha o E [0, T) E,u( , x) is de ined o almos e e y x E l[8 + and o e e y R> 0 R lim  ju( , x) 1 dx = 0 ; 0 E [0,T) El PERTURBATIONS FOR QUASILINEAR EQUATIONS  211 3) he e exis s a se E2 C [0, +oo), mes E2 = 0, such ha o x E [0, +oo) E2u( , x) is de ined o almos e e y E [0, T) and o e e y T i , 0<T, <T, and he s abili y condi ion l limo o  I u( , x) - uI ( ) I d = 0 . c xE [O,+oo) E2 Rema k 2 .1 . I u( , x) is a piecewise con inuous g .s . o he p oblem (1 .1), (1 .2) hen De ini ion 2 .1 implies (see [6]) a he line o discon inui y x = y( ) o u( , x) he Hugonio condi ion (2 .1)  y = [A( , y( ), u+ ) - A( , y( ), u-)]/(u+ - u-) (2 .2)  sign(u+ -u - ) [A( , y( ), pu - + (1 - p)u+)- - pA( , y( ), u) - (1- p)A( , y( ), u+)1 >- 0 o e e y , E (0,1) ; he e u- = u( , y - 0), u+ = u( , y + 0) . The exis ence o g .s . e he p oblem (1 .1), (1 .2) unde a ious es ic- ions en bounda y condi ions and ini ial da a was p o ed, o ins ance, in [3], [4], [10] . Theo em  2 .1 . Suppose h( , x), g( , x)  a e  measu able unc ions bounded o _< Ti, whe e TI < T is a bi a y .  Suppose w( , x) is a g .s .  o he equa ion Lw = h( , x) in Q wi h da a w(0, x) = 0, w( , 0) = wl( ) E L- ([0,T)), and ( ,x) is a g .s .  o he equa ion L = g( ,x) in Q wi h da a (0,x) = 0, ( ,0) = i ( )  E LOO ([0, T)) .  Suppose wl ( ) <_ i ( ) almos e e ywhe e in [0, T) and h( , x) <_ g( , x) almos e e ywhe e in Q . Then w( , x) < ( , x) almos e e ywhe e in Q . Fo he p oo o his heo em simila me hods o hose o pape s [2], [10] a e used . The uniqueness o he g .s . o p oblem (1 .1), (1 .2) ollows om Theo em 2 .1 . One deno es below by u( , x) he g .s . o he p oblem (1 .1), (1 .2) wi h II ( , x) - 0 . 3 . P ocess wi h agg a a ion (The case T < +oo) Theo em 3 .1 . Suppose he ollowing condi ions hold 1) A( , x, )/ < A( , x, w)/w, 0 < < w, w E I[8 + ; 212  Y . C . RYKOV 2) A( , x, )/ <_ ao(T - )a( ), E 1[8 + , aE C' (R+ ) 1 C(R + ), ao E C([0, T)), ao > 0, a(0) = 0, a is inc easing ; 3) u, ( ) < W(1/(T - )), cp E C([1/T,+oo)), W(11T) = 0, cp is in~ c easing ; 4) o ao(s)a o W(1/s) ds < +oo . Then localiza ion in he p oblem (1 .1), (1 .2) occu s and u( , x) = 0 o x > T ao(s)a o cp(1/s) ds . P oo .. Suppose he line x = y( ) is de ined by he equa ions A( , x, ul ( ))/u l ( ),  i ul ( ) :?É0, y( ) _ { Au( ,x,0),  i u, ( ) =0 ; wi h he ini ial da um y(0) = 0 .  Le us se Al( ,x) = u, ( ) o 0 <_ x < y( ) and Al ( , x) = 0 o x > y( ) .  I is easy o see ha LA, >_ 0 when x :,A y( ) and a he line o discon inui y x = y( ) (2 .1), (2 .2) hold . Flz he , hence y <- ao(T - )a ° W( 1 /(T - )),  y( 0 ) = 0, T y ( ) <  ao(s)a o cp(1/s) ds . T- Wi h he aid o assump ion 4), he applica ion o Theo em 2 .1 gi es he equi ed esul . Rema k 3 .1 . Suppose (1 .1) has he o m .(3 .1)  u +A l (T - )P(u'') x = 0, whe e Al = cons > 0, p E R, m > 1 and ul ( ) = (T- )`-T', ca > 0 . Then Theo em 3 .1 asse s he p esence o localiza ion when p - a(m - 1) > -1 . Theo em 3 .2 . Suppose he ollowing condi ions hold 1) A( , x, )/ <_ A( , x, w)/w, 0 < <_ w, w E R + ; 2)  A, ; ( , x, ) +B ( , x, ) > bo(T - ) , E l[8 + , bo E C([0,T)), bo >_ 0 ; 3) A  ( ,x, ) < ao(T - )a( ), ao E C([O,T), ao > 0, a(0) = 0, a E C'(R+ ) n C(R+ ), a inc eases, 4) a(a3) < x(a)a(a), a E [0,1], a ER+, x E C, x(O) = 0, x in- c eases ; 5) ul( ) < ~o(1/(T - )), cp E C([1/T,+oo)), w(11T) = 0, cp in- c eases ; PERTURBATIONS FOR QUASILINEAR EQUATIONS  213 6) o g(s) ds  <  +oo, a( ) o '( ) g(s) ds  <  C  <  +oo,  E  1[8 + , C = cons > 0, g(s) = ao(s)X (exp (- Tbo(o,)do,» , , (s) , p(s) exp ( ~s bo(a) da) 1 WI( )= 1/ i 1(w) . Then localiza ion in he p oblem (1 .1), (1 .2) occu s . P oo .. Le us conside he unc ion T w0 ( , x) - ( , x) exp (-  bo(s) ds) , T- whe e ( , x) is de ined by he ela ion T- (3 .2)  0 = x + a( )  g(s) ds =- x + G( , ) . wl ( ) The equa ion G( , ) = 0 wi h espec o has wo oo s : = 0, = l(1/(T - )) . When x a ies he solu ion o (3 .2) may s op o exis i G  ( , ) = 0 . Consequen ly he se o ( , x) whe e he solu ion o (3 .2) does no exis can be desc ibed by he sys em (3 .3)  x + G( , ) = 0,  G ( , ) = 0 . Now, le us conside he unc ion y( ) de ined in he ollowing way y = A ( , y, wo( , y))/wo( , y), y(0) = 0 . Then y :~ A ( , y, wo) : :~ ao(T - )a(wo) < g(T - )a( ) . F om he sys em (3 .3) o i s solu ion x = z( ) one has : z = - G - G = - G = g(T - )a( ), so y < z and lines x = y( ) and x = z( ) do no in e sec . Suppose A2( ,x) = wo( ,x) o x < y( ) and 2( ,x) = 0 o x > y( ) . I is easy o see ha T exp ~-  bo(s) ds i Lwo > [a( )g(T - ) - a(wo)ao(T - )]I G . T- Hence wi h he aid o assump ion 4) and G >_ 0 o x < z( ) one ob ains L, 2 >_ 0 o x < y( ) . Besides, a he line x = y( ) (2 .1), (2 .2) hold . Since u(0, x) < A2 (0, x) we ha e u( , x) < A2 ( , x) in Q . 214  Y . G . RYKOV Le us ew i e (3 .2) : Hence ¡ T-  ¡ wl ( ) x + a( )  0  0 J  g(s) ds - a( ) J  g(s) ds = 0 . by i ue o assump ion 6) . When x is su icien ly la ge he e is no solu ion o (3 .2) and z(T - 0) < +oo . This ends he p oo . Co olla y 3 . l . I in addi ion o assump ions o Theo em 3 .2 he ol- lowing inequali y holds o ixed x 7~ 0 and => T - 0 . o T- 0 x + a( )  g(s) ds <_ C= cons wl( ) a( )  g(s) ds < l( ),  E R + , 0 whe e 77( ) dec eases, l(+oo) = 0 hen u( , x) is bounded as => T- 0 o e e y ixed x :y~ 0 . P oo . Indeed, om (3 .2) we ha e ¡ wl ( )  ¡ T- l( ) > a( ) J  g(s) ds = x + a( ) J  g(s) ds > x, 0  o o < l -1 (x) . Since u( , x) < A2( , x) one ge s he boundedness u( , x) Rema k 3 .2 . Fo he equa ion (3 .1) Theo em 3 .2 gi es he localiza- ion p esence when p - a(m - 1) >_ -1, while Co olla y 3 .1 gi es he boundedness o g .s . o x 7~ 0 and =~> T - 0 when p - a(m - 1) > -1 . Theo em 3 .3 . Suppose he ollowing condi ions hold : 1) A( , x, )/ < A( , x, w)/w, 0 < <_ w, w E R + ; 2) A,, ( , x, )  >_  ao(T - )a( ),  E  l[8 + , ao  E  C([O, T)), a 0  >_ 0, a(0) = 0, a E C l (I[8 + ) n C(& + ), a inc eases ; 3) 5 2ao (T - )a( ) _< A( , x, )/ <_ b l a o (T - )a( ), E I[8 + , 0 < 62 <5 1 <1 ; 4) px(a)a(a) ? a(a l) >_ X(n)a(0), w >_ 1, 61p < 1, aE [0,1], a E R+, X E C([0,1]), X(0) = 0, X inc eases ; 5) B( ,x, )+A .,( ,x, ) <_ b o (T- ) , E I[8 + , b o E C([O,T]), b o >_ 0 ; 6) u, ( ) > cp(1/(T - )), cp E C([1/T,+oo)), w(11T) = 0, cp in- c eases PERTURBATIONS FOR QUASILINEAR EQUATIONS  215 7) g(s)s < 01(s) o g(o , ) da, 0 < s < T ; cp'(s)s > 02(s) ;p(s), s > 1/T ; sa'(s) > 03(s)a(s), s > 0;'04(s) = bo(1/s)/s + 02(s), whe e o ¡ (s) (i = 1, 2, 3) a e mono onic (in pa icula may be cons an e) and 1 - Y'1 0 wl( )/[03( )04(1/wl( ))] > M( ), E R + , pE C(R + ), > 0, l # 0, M, does no inc ease ; 8 ) o H(s)ds=+oo, H(s)-ao(s)a(exp( T bo(a)da) -1 ( o g(a)lo)) V(V) - m( ) o 1( ) g(s) ds, e = cons > 0, o g(s) ds < +oo . Then he e is no localiza ion in he p oblem (1 .1), (1 .2) and u( , x) > 0 o 0 < x < 52 T H(a) da . P oo . Le us conside he unc ion wo( , x) in oduced in he p oo o Theo em 3 .2 . Suppose y( ) is de ined by he equa ion y = A( , y, wo( , y))/wo( , y) wi h he ini ial da um y(0) = 0 . By analogy wi h he p oo o Theo em 3 .2 one s a es ha he cu e x = y( ) is con ained in he domain o exis en e o he solu ion o equa ion (3 .2) . Le us ega d he same compa ison unc ion 2( ,x) as in he p oo o Theo em 3 .2 . As G,> 0 o x < z( ) one has Lwo < 0 o x < y( ) and U ( , x) > >12 ( , x) in Q . Now he equa ion G( , ) = 0 has wo oo s and he oo o he equa- ion G = 0 lies be ween hem by i ue o Rolle's Theo em . Conse- quen ly he solu ion > 0 o he equa ion (3 .2) wi h ixed x, always exceeds he solu ion o he equa ion G = 0 wi h he same ixed . Hence T- wi( ) 0 =G  =á ( ) (1  g(s) ds -  g(s) ds) + 0  0 o + a( )g o wl( )( 1 1) I ( )wl( ) 2 , T-  wi ( ) g(s) ds = g(s) ds - a( ) g ° wl( )( i 1y(V)W1(V)2 . o  o  a '( ) Using condi ions 7) one es ima es : T- g(s) ds > wi ( ) 9(s) ds [ 1 - a'( ) i o ( 1) ( ) 1  wl( )] , T s i(s) = exp (£  bo(o , ) do , ) [s -1 bo( 1 /s)cp(s) + sw (s)] ? VI(S)04(8) ; lis l T-  wl( )  a( )  P1 ow, ( ) g(s) ds > ~  g(s) ds [ 1 - a(V) 04( 1 /W1( ))] wi( )  '~~  wi( )  g(s) ds ~1 -  Y51 0 wl( )  g (s) dsp( )= ( ) . o  03( )V)4( 1 /wl( )) o 21 6  Y . G . RYKOV Since p(s) and w1(s) do no inc ease (s) does no inc ease oo . Consequen ly ( , x) > -1 ( o - g(s) ds) , hence T  T- wo( , x) > exp ~-  bo(s) ds  -1  g(s) ds  - H1 (T - ) . T-  )  ( 0 Fu he , y = A( , y, w 0 )/w 0 ? 62ao(T - )a(wo) T  I'- >_ S2ao(T- )a (exp (~  bo(s) d) s -1 (0 g(s) ds  =6 2 H(T- ) . T- This inequali y implies y( ) > 52 T H(u) _  du and we ob ain he e- qui ed esul wi h he aid o assump ion 8) . Co olla y 3 .2 . Suppose condi ions 1)-7) o he Theo em 3 .3 hold, bu ins ead o condi ion 8) assume lim H 1 (T - ) = +oo . Suppose u( , xo) > =>T 0 o some xo and close o T . Then u( , x0) unbounded as => T - 0 . P oo .. In he p oo o Theo em 3 .3 we had he es ima e w 0 ( , x) >_ H 1 (T - ) . Since u( , x) >_ wo( , x) o x < y( ), he asse ion o he co olla y is ue . Rema k 3 .3 . Fo he equa ion (3 .1) Theo em 3 .3 asse s he local- iza ion absence when p- ca(m - 1) < -1, p > -1 .  Indeed, in his case ao(s) = A1sP, bo(s)  0, X(s) = s m-1 , a(s) = msm -1 , g(s) _ A l sP, 01 (s)  p + 1, 02 (s)  a, Y'4(8) = 02(s), 03 (s) = m - 1, P(s) 1 - (p + 1)/(a(m - 1)), (s) = A1 [a(m - 1) - p - 1] (s + T-«)-(P+1)/«, a(m - 1) (p + 1) H(s) = mA1sP I s-« (  a(m - 1)  ) -  - T-« a(m-1)-p-1 I ollows om Co olla y 3 .2 ha u( , x) is unbounded as => T - 0 and x ixed, since -«  n ( m - 1)  1 - «/(P+1) H l (S) - s  la(m  1)  1 Suppose p- a(m - 1) = -1 . Then Theo em 3 .3 is in alid because o assump ion 8) . Bu one can choose 02(8) = as«/(s« -T - «), (s) = A1(p+ 1)-1T-«(s+T-«)-m, H1(s) = T-«[(T/s)«(m-1)/m - l] . The unboundedness o u( , x) as => T - 0 and x is no oo la ge ollows om Co olla y 3 .2 . PERTURBATIONS FOR QUASILINEAR EQUATIONS  217 Theo em 3 .4 . Suppose condi ions 1)-6) o Theo em 3 .3 hold and o g(s) ds = +oo . Then he e is no localiza ion in he p oblem (1 .1), (1 .2) and u ( , x) > 0 o 0 < x < cons ( T g(s) ds)62 . P oo . Le us conside he unc ion A2( , x) de ined in he p oo o Theo em 3 .2 . We ha e y = A( , y, w 0 )/w0, y(0) = 0 . The solu ion o his Cauchy p oblem is no iden ically ze o since A( , 0, wo) # 0 by assump ion . Hence, he e exis such x* > 0, * > 0 ha y( *) = x* . F u he , o > * one ob ains y > 62ao(T - )a(wo) > 62g(T - )a( ) . I is ob ious ha wl ( ) < T by he de ini ion o unc ion w, ( ) . So we ha e om (3 .2) Now, y = a( ) -1 g(s) ds < a( ) T  g (s) ds T  T- 7,  1 - 1 y > 62yg(T - ) ~  g(s) ds J , T- Y( ) > x*  ¡T  g(s) ds1 -bz ~ J ¡7 .  g(s) ds 16z ~ J  => +oo T- *  T- J as => T - 0 . This ends he p oo . y( *) = Rema k 3 .4 . In he case o equa ion (3 .1) Theo em 3 .4 s a es he absence o localiza ion o p < -1 . Theo em 3 .5 . Suppose assump ions 1)-6) o Theo em 3 .3 hold and o g(s) ds = +oo . Suppose he ollowing condi ions hold : 1) , T g(T) dT < sg(s)O1(s), 0 < s < T ; W'(s)s > cp(s)02(S), s > 1/T ; sa'(s) > a(s)03(s), s > 0 ; 04(s) - bo(1/s)/s+02(s),whe e o¡ (s) (i = 1, 2, 3) a e mono onic unc ions (in pa icula may be cons an e) and 0 1 o wl( ) +  3( ) 4(l/wl( )) < M( ), E 1I8 + , M E lPlP C(R + ), , > 0, l 9~ 0, i inc eases ; 2) ( ) - g(wi( ))w,( )p( ) =* +oo as = :> +oo ; 3) H2 (s) - exp (- s bo(o-) do) -i ( s g (u) do , ==> +oo as s +0 .